light-matter interaction : physics and engineering at the nanoscale

LIGHT–MATTER INTERACTION

Light–Matter InteractionPhysics and Engineering at the Nanoscale

Second edition

John WeinerUniversité Paul Sabatier, Toulouse France and IFSC Universidade des Sao Paulo,

Sao Carlos, SP Brazil

Frederico NunesFederal University of Pernambuco, Recife, Brazil

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Preface

Light–matter interaction pervades the disciplines of optical and atomic physics, con-densed matter physics, electrical engineering, molecular biology, and medicine withfrequency and length scales extending over many orders of magnitude. Deep earth andsea communications use frequencies of a few tens of Hz, and X-ray imaging requiressources oscillating at hundreds of petaHz (1015 s–1). Length scales range from thousandsof kilometres to a few hundred picometres. Although the present book makes no pre-tence to offer an exhaustive treatise on this vast subject, it does aim to provide advancedundergraduates, graduate students, and researchers from diverse disciplines, the princi-pal tools required to understand and contribute to rapidly advancing developments inlight–matter interaction centred at optical frequencies and length scales, from a few hun-dred nanometres to a few hundredths of a nanometre. Classical electrodynamics, withan emphasis on the macroscopic expression of Maxwell’s equations, physical optics, andquantum mechanics provide their own perspectives and physical interpretations at theselength scales. Circuit theory and waveguide theory from electrical engineering furnishuseful analogies and often offer important insights into the nature of these interactions.A principal aim of this book is to deploy this arsenal of powerful tools so as to render thesubject in forms not likely to be encountered in standard physics or engineering courses,while not straying too far into eccentricity.

This book builds on an earlier one, Light–Matter Interaction, Physics and Engineeringat the Nanoscale, that I wrote with Frederico Nunes. Much of the material in Chapters 2,4–8, and 12 remains essentially unchanged from this earlier work, although the organ-isation has been improved and some minor mistakes corrected. Chapters 1 and 3 havebeen expanded somewhat, and chapters 9–11 are entirely new. The motivation behindwriting this book was to include the subject matter of current research interest, such asmetamaterials and light forces on atomic and nanoscale objects. The chapter on mo-mentum in fields and matter grew out of the considerations on light forces, and I wassurprised to find how subtle and slippery this subject is. It is my hope that the chapterwill provide a deeper understanding of how momentum transport can affect the natureof forces and force distributions on ponderable objects. The subject of momentum fluxbetween fields and matter is as important as the more frequently treated (via Poynting’stheorem) subject of energy flux.

After a historical synopsis of the major milestones in the human understanding oflight and matter in Chapter 1, the subject begins in earnest with a review of conven-tional electrodynamics in Chapter 2, Elements of Classical Electrodynamics. The intent isto reacquaint the reader with electric and magnetic force fields and their interactionswith ponderable media through Maxwell’s equations and accompanying force laws,such as the common Lorentz force law. We emphasise here, macroscopic quantities

vi Preface

of permittivity and permeability, and through the constitutive relations, polarisationand magnetisation fields. Dipole radiation, space-propagating and surface-propagatingwave solutions to Maxwell’s equations are all fundamental to understanding energy andmomentum transport around, and through, atomic-scale and nanoscale structured ma-terials. The chapter ends with a development of plane wave propagation in homogenousmedia and at dielectric and metallic surfaces.

Chapter 3, Physical Optics of Plane Waves, introduces the phasor representation, aswell as the first mention of the expressions for energy density and energy flux. Thesekey notions will recur continually throughout the book in various contexts. The secondhalf of the chapter treats reflection and transmission, total internal reflection, the Fresnelcoefficients, real material interfaces, and plane-wave behaviour in a lossy, conductivemedium at high frequency.

Chapter 4, Energy flow in Polarisable Matter, covers the time evolution of energyflux when electromagnetic waves propagate through media with electric polarisation.We point out analogies between the behaviour of classical fields in bulk matter withthe energy dynamics of reactive and dissipative electronic circuits. In the section onpolarisation and polarisability, it is shown how the macroscopic electric, polarisation,and displacement fields can be related to microscopic atomic and molecular propertiesthrough the Clausius–Mossotti equation that expresses the dielectric constant of a mater-ial (a macroscopic property) in terms of the microscopic polarisability of the constituentatoms or molecules.

Chapter 5, The Classical Charged Oscillator and the Dipole Antenna, is next presentedfor its own intrinsic and practical interest as well as an application of the foregoing prin-ciples. It is shown how a ‘real’ antenna can be built up from an array of oscillating chargesand how an array of macro-antennas can be used to concentrate the spatial direction ofemission or reception. The treatment here is fundamental with a fairly conventionalengineering perspective, but it lays the groundwork for a thorough understanding ofatomic, molecular, and nanoscale dipole emitters and absorbers.

Chapter 6, Black-body Radiation, reviews the Rayleigh–Jeans and Planck distributions.The presentation shows how any radiation law must be the product of mode count-ing and the distribution of energy per mode. It is shown that the key to avoiding the‘ultraviolet catastrophe’, and to obtaining agreement with experimental measurement, isto use the Planck distribution. This chapter also provides some necessary backgroundmaterial and context for the discussion of dipole emitters interacting with hyperbolicmetamaterials.

Chapter 7, Surface Waves, is devoted to a fairly extensive discussion of waves at theinterface between dielectrics and metals, because they play such an important role in‘plasmonic’ structures and devices. In fact, this propagation can be expressed in termsof circuit and waveguide theory, familiar to electrical engineers. At the opening of thetwentieth century, surface waves were thought to be the means by which radio trans-mission was carried beyond the earth’s curvature, and the importance of this subjectmotivated the extensive analysis that appears in Arnold Sommerfeld’s celebrated Seriesof Lectures on Theoretical Physics, Volume VI ; especially chapter VI, Problems of Radio. Al-though the ionosphere was found to be responsible for long-distance radio transmission,

Preface vii

Sommerfeld’s analysis laid the groundwork for understanding atomic and molecularemission near surfaces and the importance of anisotropic metamaterials for reflectionand transmission.

Chapter 8, Transmission Lines and Waveguides, establishes the correspondence be-tween classical electromagnetics and circuit properties such as capacitance, inductance,and impedance. Rectangular and cylindrical geometries are discussed at length becauseof their importance in conventional microwave-scale waveguides as well as in nano-fabricated light-guiding devices. TM and TE waveguide modes (as distinct from TMand TE polarisation) are discussed in detail. The chapter ends with a presentation ofhow waveguide modal analysis and impedance matching can be used to guide the designof nanoscale optical devices.

Chapter 9 introduces the notions of ‘left-handed materials’, negative-index metama-terials, and waveguides, and how they may be used to tailor light flow. The field ofmetamaterials develops new directions and applications with the appearance of eachmonthly, or even bi-weekly, issue of the principal research journals. To try to presentthe ‘latest and greatest’ in this chapter would be futile, so the emphasis is rather onthe basic physics, and especially, transmission and reflection in periodic stacked layers.This geometry is the simplest implementation of fabricated anisotropic materials withengineered properties of transmission and reflection.

Chapter 10 examines the meaning of momentum in electromagnetic fields and howthat momentum interacts with ponderable media. Energy conservation in electromag-netics enters by way of Poynting’s theorem, and the Poynting vector expresses energypower flux (Watts per m2 in SI units). The Einstein thought experiment establishes theneed for a similar conservation principle for momentum transmission between fields andmatter. The question of field momentum is crucial to a thorough understanding of lightforces (which must be equivalent to the time rate of change of momentum as it passesbetween field and object), such as the radiation pressure force and the dipole-gradientforce. We examine the Abraham-Minkowski controversy on the ‘correct’ way to expressoptical momentum inside ponderable matter and discuss, in some detail, the key experi-ments whose motivation was to resolve the controversy. The experiments, at least at thiswriting, have only managed to send the conflicting analyses in new directions. The chap-ter ends with an extended discussion of light momentum on a point dipole (standing infor a two-level atom) and summarises important articles cited and referenced in Chapter10. This discussion is a natural lead-in to the next chapter on atom-optical forces, opticalcooling, and trapping.

Chapter 11 presents the simplest and most intuitive approach to atom-light-fieldinteractions: the atom as a damped harmonic oscillator with spontaneous emission asthe damping agent. The next step is the semiclassical two-level atom, initially intro-duced as a point dipole (but now with two internal states) at the end of Chapter 10.The semiclassical two-level atom sets the stage for establishing light forces at the atomiclevel: the dipole-gradient force and the radiation pressure force. Finally, the optical Blochequations are introduced, which facilitates the presentation of the last section on atomDoppler cooling.

viii Preface

Chapter 12, Radiation in Classical and Quantal Atoms, introduces light–matter inter-action at the atomic scale (a few hundred picometres) and at interaction energies lessthan, or comparable to, the chemical bond. Under these conditions the subject can bevery well understood through a semiclassical approach in which the light field is treatedclassically and the atom quantally. We therefore retain the classical electrodynamicstreatment while presenting a very simple quantum atomic structure with dipole tran-sitions among atomic and molecular internal states. We take a physically intuitive, wavemechanical approach to the quantum description in order to bring out the analogies be-tween classical light waves, quantum matter waves, classical dipole radiation, and atomicradiative emission.

A number of Appendices have been included that provide supplementary discus-sion of the analytical tools used to develop the physics and engineering of light–matterinteraction. Appendix A lists numerical values of important fundamental constants anddimensions of electromagnetic quantities. Appendix B is a brief discussion of systemsof units in electricity and magnetism. Although the Système International (SI) has nowbeen almost universally adopted, it is still worthwhile to understand how this system isrelated to others; what quantities and units can be chosen for ‘convenience’ and what arethe universal constraints that all systems must respect. Students should not be deterredfrom studying earlier articles and texts simply because of an unfamiliar system of units.Appendix C is a brief review of vector calculus that readers have probably already seen,but who might find a little refresher discussion useful. Appendix D discusses how theimportant differential operations of vector calculus can be recast in different coordinatesystems. Although the Cartesian system is usually the most familiar, spherical and cy-lindrical coordinates are practically indispensable for frequently encountered problems.Much of the book deals with harmonically oscillating fields, and Appendix E is a suc-cinct review of the quite useful phasor representation of these fields. Finally, AppendicesF, G, and H present the properties of the special functions, Laguerre, Legendre, andHermite, respectively, that are so commonly encountered in electrodynamics and quan-tum mechanics. These Appendices are an integral part of the book, not just some ‘boilerplate’ nailed on at the end. Readers are strongly encouraged to pay as much attention tothem as they do to the Chapters.

Most of the material in this book is not new nor original with the authors. Excellenttexts and treatises on classical electrodynamics, physical optics, circuit theory, wave-guide and transmission line engineering, atomic physics, and spectroscopy are readilyavailable. The real aim of this book is take the useful elements from these disciplines andto organise them into a course of study applicable to light–matter interaction at the nano-scale and the atomic scale. To the extent, for example, that waveguide mode analysis andsound design practice in microwave propagation inform the nature of light transmissionaround and through fabricated nanostructures, they are relevant to the purposes of thisbook. Rugged, reliable coherent light sources in the optical and near-infrared regime,together with modern fabrication technologies at the nanoscale, have opened a new areaof light–matter interaction to be explored. This exploration is far from complete, but thepresent book is intended to serve as a point of entry and a useful account of some of theprincipal features of this new terrain.

Fundamental Constants and Symbols

mu atomic mass constant 1.660539× 10–27 kgmp proton mass 1.672622× 10–27 kgme electron mass 9.109382× 10–31 kgg acceleration of gravity 9.80665 m s–2

G gravitation constant 6.674287× 10–11 m3 kg–1 s–2

re classical electron radius 2.817941× 10–15 mF force NG momentum kgm s–1

ε0 vacuum permittivity 8.854187× 10–12 Fm–1

μ0 vacuum permeability 12.566370× 10–7 NA–2

ε, εr dielectric constant unitlessμ,μr relative permeability unitlessh Planck constant 6.626070× 10–34 J sh Planck constant/2π 1.054572× 10–34 J sc vacuum speed of light 299792458 m s–1

ν frequency s–1

ω angular frequency s–1

λ wavelength mk wave vector m–1

T temperature KkB Boltzmann constant 1.380650× 10–23 JK–1

σ Stefan–Boltzmann constant 5.670367× 10–8 Wm–2 K–4

NA Avogadro constant 6.022142× 1023 mol–1

R molar gas constant 8.314472 Jmol–1 K–1

μB Bohr magneton 927.400999× 10–26 JT–1

e electron charge 1.602177× 10–19 Cq electric charge Cρ electric charge density Cm–3

magnetic flux WbE electric field Vm–1

B magnetic induction field Wbm–2

D electric displacement field Cm–2

H magnetic field strength Am–1

J electric current density Am–2

P electric polarisation field Cm–2

M magnetisation field Wbm–2

S power flux density Wm–2

x Fundamental Constants and Symbols

χ ,χe electric susceptibility unitlessχm magnetic susceptibility unitlessL inductance WbA–1

C capacitance FE,E energy JP power WI electric current C s–1

R electric resistance �

V voltage, potential J C–1

ρ resistivity �mσ , κ conductivity Sm–1

Z impedance �

Y admittance Sn refractive index unitlessη Re[n] unitlessκ Im[n] unitlessR reflection coefficient unitlessT transmission coefficient unitless

Contents

1 Historical Synopsis of Light–Matter Interaction 1

1.1 Light and matter in antiquity 11.2 The Golden Age of Sciences in Islam 21.3 Light and matter in the European Renaissance 31.4 The revolution accelerates 71.5 One scientific revolution spawns another 91.6 Summary 111.7 Further reading 12

2 Elements of Classical Electrodynamics 13

2.1 Introduction 132.2 Relations among classical field quantities 132.3 Classical fields in matter 152.4 Maxwell’s equations 162.5 Static fields, potentials, and energy 182.6 Three examples of problem solving in electrostatics 222.7 Dynamic fields and potentials 342.8 Dipole radiation 382.9 Light propagation in dielectric and conducting media 41

2.10 Summary 452.11 Exercises 452.12 Further reading 47

3 Physical Optics of Plane Waves 48

3.1 Plane electromagnetic waves 483.2 Plane wave reflection and refraction 593.3 Summary 853.4 Further reading 85

4 Energy Flow in Polarisable Matter 86

4.1 Poynting’s theorem in polarisable material 864.2 Harmonically driven polarisation field 874.3 Drude–Lorentz dispersion 884.4 Polarisation from polarisability 954.5 Summary 994.6 Further reading 99

xii Contents

5 The Classical Charged Oscillator and the Dipole Antenna 100

5.1 Introduction 1005.2 The proto-antenna 1005.3 Real antennas 1025.4 Summary 1065.5 Further reading 107

6 Classical Black-body Radiation 108

6.1 Field modes in a cavity 1086.2 Planck mode distribution 1116.3 The Einstein A and B coefficients 1126.4 Summary 1146.5 Further reading 114

7 Surface Waves 115

7.1 Introduction 1157.2 History of electromagnetic surface waves 1157.3 Plasmon surface waves at optical frequencies 1167.4 Plasmon surface wave dispersion 1267.5 Energy flux and density at the boundary 1347.6 Plasmon surface waves and waveguides 1387.7 Surface waves at a dielectric interface 1427.8 Summary 1517.9 Exercises 152

7.10 Further reading 153

8 Transmission Lines and Waveguides 154

8.1 Introduction 1548.2 Elements of conventional circuit theory 1548.3 Transmission lines 1598.4 Special termination cases 1688.5 Waveguides 1728.6 Rectangular waveguides 1788.7 Cylindrical waveguides 1828.8 Networks of transmission lines and waveguides 1898.9 Nanostructures and equivalent circuits 198


9 Metamaterials 211

9.1 Introduction 2119.2 Left-handed materials 2119.3 Negative index metamaterials and waveguides 213

Contents xiii

9.4 Reflection and transmission in stacked layers 2169.5 Summary 2539.6 Bibliography 253

10 Momentum in Fields and Matter 256

10.1 Introduction 25610.2 Einstein Box thought experiment 25710.3 Balazs thought experiment 25910.4 Field equations and force laws 26410.5 Summary 29710.6 Further reading 29810.7 Bibliography 298

11 Atom-Light Forces 301

11.1 Introduction 30111.2 The atom as a damped harmonic oscillator 30211.3 Radiative damping and electron scattering 30511.4 The semiclassical two-level atom 30611.5 The dipole-gradient and radiation pressure forces 31711.6 Summary 32611.7 Exercises 32611.8 Further reading 326

12 Radiation in Classical and Quantal Atoms 328

12.1 Introduction 32812.2 Dipole emission of an atomic electron 32812.3 Radiative damping and electron scattering 33112.4 The Schrödinger equation for the hydrogen atom 33212.5 State energy and angular momentum 34212.6 Real orbitals 34512.7 Interaction of light with the hydrogen atom 34612.8 The fourth quantum number: intrinsic spin 35812.9 Other simple quantum dipolar systems 358


Appendices

Appendix A Numerical Constants and Dimensions 367

A.1 Numerical values of some fundamental constants 367A.2 Dimensions of electromagnetic quantities 368

xiv Contents

Appendix B Systems of Units in Electromagnetism 369

B.1 General discussion of units and dimensions 369B.2 Coulomb’s law 370B.3 Ampère’s law 372

Appendix C Review of Vector Calculus 375

C.1 Vectors 375C.2 Axioms of vector addition and scalar multiplication 377C.3 Vector multiplication 378C.4 Vector fields 381C.5 Integral theorems for vector fields 385C.6 Useful identities of vector calculus 387

Appendix D Gradient, Divergence, and Curl in Cylindricaland Polar Coordinates 388

D.1 The gradient in curvilinear coordinates 391D.2 The divergence in curvilinear coordinates 391D.3 The curl in curvilinear coordinates 392D.4 Expressions for grad, div, curl in cylindrical and polar coordinates 393

Appendix E Properties of Phasors 396

E.1 Introduction 396E.2 Application of phasors to circuit analysis 397

Appendix F Properties of the Laguerre Functions 400

F.1 Generating function and recursion relations 400F.2 Orthogonality and normalisation 401F.3 Associated Laguerre polynomials 401

Appendix G Properties of the Legendre Functions 404

G.1 Generating function 404G.2 Recurrence relations 405G.3 Parity 406G.4 Orthogonality and normalisation 407

Appendix H Properties of the Hermite Polynomials 409

H.1 Generating function and recurrence relations 409H.2 Orthogonality and normalisation 410

Index 413

Authors 417

1

Historical Synopsis of Light–MatterInteraction

The phrase ‘light–matter interaction’ covers a vast realm of physical phenomena fromclassical to quantum electrodynamics, from black holes and neutron stars to mesoscopicplasmonics, and nanophotonics to subatomic quantum objects. The term ‘interaction’implies that light and matter are distinct entities that influence one another through someintermediate agent. The history of scientific inquiry from the earliest times to the presentday might be neatly summarised into three questions: what is the nature of light itself,of matter itself, and of the interaction agent? We now know from Einstein’s celebratedequation E = mc2 that light (E) and matter (m) are fundamentally manifestations of thesame ‘thing’, related by a universal proportionality constant: the square of the speed oflight in vacuum (c2). Nevertheless, under ambient physical conditions normally found onearth, the distinction between light and matter makes sense; their interaction meaningfuland worth studying.

1.1 Light and matter in antiquity

In the fifth century BC, Leucippus, a Greek philosopher from Miletus (now in Turkey),founded the school of atomism in which the universe is composed of immutable, in-destructible, indivisible atoms and the space through which they move: the void. Hisbest student was Democritus (460–370 BC) who elaborated the atomistic construct ofthe universe, attributing natural phenomena to the motion of atoms and the diversityof material objects to their shapes and interlocking structures. The most extensive ac-count of the Leucippus-Democritus atomic theory appears in an extended epic poem,De rerum natura (The Nature of Things) by Lucretius, a Roman, who lived much later(99–55 BC).

A contemporary of Democritus, the Greek philosopher Empedocles (490–430 BC),proposed that the cosmos was composed of four elements: fire, air, water, and earth.Like the atomist school, these elements were immutable and the diversity of nature arose

Light-Matter Interaction. Second Edition. John Weiner and Frederico Nunes.© John Weiner and Frederico Nunes 2017. Published 2017 by Oxford University Press.

2 Historical Synopsis of Light–Matter Interaction

Empedocles cosmic cycle

Presence of life

Presence of life

Life not present Life not present

ContentionbetweenLOVE

and STRIFE

ContentionbetweenSTRIFE

and LOVE

Puredomain

of STRIFE:KAOS

Puredomain

of LOVE:HARMONY

Figure 1.1 The cosmic cycle of Empedocles. Creative Commons by-sa 3.0,Paolo Anghileri.

from their combinations. The dynamics of the combinations are affected by two forces,repulsive and attractive, known as love and strife, respectively. Figure 1.1 illustratesEmpedocles’ scheme.

Empedocles is also credited with proposing the first theory of light. His idea was thatlight particles stream out of the eyes and contact material objects. Euclid (about 300 BC)assumed this flux moved in straight lines and used the idea to explain some opticalphenomena in Euclid’s Optics, a very influential early treatise on optics. Euclid’s Optics,in turn, influenced Claudius Ptolemy (AD 90–168), a Roman citizen living in Egypt,whose writing on geocentric astronomy was considered definitive until the EuropeanRenaissance.

1.2 The Golden Age of Sciences in Islam

The Golden Age of Sciences in Islam was around the year AD 1000, at the time of Ibn-i-Sina (Avecenna), the last of the Islamic mediaevalists, and Ibn-al-Haitham (Alhazen,AD 965–1039), the first of the modernists. Alhazen enunciated that a ray of light, whenpassing through a medium, takes a path that is easier and ‘quicker’, anticipating Fermat’sprinciple of least time by many centuries (see Section 1.3). Contrary to Empedocles andPtolomy, Alhazen believed that the eye detected light from an external source. Figure 1.2shows an illustration from the title page of a Latin translation of Alhazen’s Book ofOptics.

Light and matter in the European Renaissance 3

Figure 1.2 Illustration from the title page of the 1572 edition of OpticaeThesarus–Latin translation of Alhazen’s Book of Optics. The figure illustratesmany of the properties and uses of light. Perspective, refraction, reflection, therainbow, the periscope, and early photonic naval defences are represented. Figure inpublic domain.

1.3 Light and matter in the European Renaissance

By the beginning of the seventeenth century, the certitude of received ideas was crum-bling. Earth as the centre of the universe and Europe as the centre of the earth wascast into doubt. The Americas were discovered by European explorers between 1492and 1504, the earth had been circumnavigated by 1522, and the Ptolemaic geocentricastronomical system had been effectively overthrown by the Copernican heliocentricrevolution of 1543. The invention of the telescope in 1608 enabled Galileo to show


that Jupiter’s moons revolved around that planet, not the earth. This discovery ofobservational astronomy ran counter to Church dogma, but Pope Urban VIII was actu-ally sympathetic to Galileo’s scientific way of thinking. Unfortunately, Galileo publisheda ‘dialogue’ consisting of a conversation among three people: Salviati, a smart scientistwho bore a striking resemblance to Galileo and who argued for the Copernican system;Sagredo, a sort of moderator in the dialogue who asked intelligent questions; and Sim-plicio, who tried to defend the conventional Aristotelian method of speculating by purethought before eventually having recourse to cite the mysteries of the unknowable handof God. Unfortunately, Simplicio’s arguments uncomfortably paralleled those of UrbanVIII himself. The Pope did not take kindly to being embarrassed and Galileo lost hisprotection from the Inquisition. Figure 1.3 shows the frontispiece and title page of theDialogue published in 1632.

Into this fluid situation stepped René Descartes (1596–1650) with a new world view.Descartes proposed that the universe consisted only of matter and motion. Forces couldonly be propagated among massive bodies by actual contact, and therefore the apparentspace between celestial bodies, the ‘void’ of Democritus, was actually filled with a kindof very fine-grained material medium or plenum. Light emission, reflection, refraction,

Figure 1.3 Frontispiece and title page of the Dialogue published in Florence in 1632. Simplicio is onthe left. Figure in public domain.

Light and matter in the European Renaissance 5

and absorption, were all explained in terms of material flux. The notion of force ‘fields’and action at a distance had no place in the Cartesian system of the universe. Des-cartes’ interpretation of refraction, however, was severely challenged by Pierre de Fermat(1601–1665) who explained the deviation of light rays on the basis of the principle of leasttime. Applying this principle, Fermat derived that the sines of the incident and refractedangles are in constant ratio, essentially the equivalent of what we now commonly term‘Snell’s law’. Descartes also derived this law, but his interpretation of light as particleflux required greater velocity in the denser medium, whereas the principle of least timeimposed a slower velocity. Fermat’s principle is in accord with the modern expressionfor the velocity of light, v = c/n where n, the index of refraction, is unity in free spaceand greater than unity in material media.

The next significant observation was light ‘diffraction’, a term coined by FrancescoGrimaldi (1618–1663) to describe the appearance of light beyond the geometricalshadow boundary defined by the supposed rectilinear motion of light-particle flux. Fig-ure 1.4 shows a portrait of Grimaldi and his diagram for light appearing beyond thegeometric limits of an angular opening. Diffraction was also observed by Robert Hooke(1635–1703) who conjectured that light was due to rapid vibratory motion of the verysmall particles of which ordinary matter is composed. Furthermore Hooke had the bril-liant insight that light (still considered as a kind of matter flux) propagated outwardsfrom the centre of each tiny vibrating centre in circular figures and that light ‘rays’ weretrajectories at right angles to these circular figures. This view of light propagation laidthe foundation for the construction of wave fronts from which Hooke was able to ex-plain refraction. He also tried to interpret colours in terms of refraction, but his colourtheory was challenged by Isaac Newton (1642–1727) who correctly interpreted colour

A BC D

G HE F

I N L M O K

Figure 1.4 Grimaldi and his diffraction diagram that shows that the rays I,Kextend beyond the angular limits N,O defined by the two apertures. Figures inpublic domain.


as an intrinsic property of light and not a distortion of it due to refraction. AlthoughHooke took the first steps toward a wave theory of light, it was Christiaan Huygens(1629–1695) who put it on a firmer foundation by expressing refraction in terms ofthe principle that each element of a wavefront may be regarded as the centre of a secondarydisturbance giving rise to spherical waves. The wavefront at any later time is the envelope ofall such secondary wavelets. This principle was later refined and extended by the Frenchengineer Augustin-Jean Fresnel (1788–1827) to establish modern wave optics, based onthe Huygens-Fresnel principle. It successfully explains light intensity modulations dueto diffraction.

Throughout antiquity and into the seventeenth century, however, light and matterwere not considered to be intrinsically different. Light was simply a manifestation of mat-ter, either in linear flux, or in vibratory motion. Huygens also discussed the phenomenonof double diffraction in Iceland crystal and interpreted it in terms of the propagationof ordinary and ‘extraordinary’ waves within the crystal. These studies were later con-sidered by Newton and led to the discovery of light polarisation. Newton believed thatlongitudinal, compressive waves could never account for polarisation and his argumentsessentially laid to rest the wave interpretation of light until it was revived by ThomasYoung (1773–1829) who demonstrated interference in the celebrated Young’s double slitexperiment. Figure 1.5 shows the classic diagram of the two-slit experiment that providedpowerful evidence of the wave nature of light.

Meanwhile, the atom theory of matter was being transformed from an antiquespeculative philosophical proposition to a working scientific hypothesis that was re-fined, enlarged, and tested by quantitative experiments. In 1643, Evangelista Torricelli(1608–1647) established that the ‘air’ of everyday experience exerted pressure andwas therefore a tangible, if rarified, material composed of particles in constant motion.

P

x

θ

Figure 1.5 Thomas Young and a diagram of the two-slit experiment. Noticethat a bright band always appears on the centre-line. Left figure created underCreative Commons by-sa 3.0; right figure in public domain.

The revolution accelerates 7

In England, Newton published the Philosophiae Naturalis Principia Mathematica in 1687,laying out his laws of motion and theory of gravitation supported by precise, quantitativecelestial mechanics. Although Newton did not like to admit it, his gravitational force lawdid not require an intervening plenum between bodies with mass. Contrary to Descartes,action-at-a-distance was tacitly, if not overtly, admitted.

Robert Boyle (1627–1691) established one of the fundamental gas laws—that thepressure was inversely proportional to volume at fixed temperature—and helped thescience of chemistry emerge from obscurantist alchemy by publishing a truly scien-tific treatise: The Sceptical Chymist. His distinction between heterogeneous mixtures andhomogenous compounds laid the groundwork for the truly revolutionary advances ofthe eighteenth century.

1.4 The revolution accelerates

At the beginning of the eighteenth century the nature of light was still very much indoubt. The notion, commonly attributed to Newton, that light consisted of beams ofparticles, or ‘corpuscles,’ travelling in straight lines through homogenous media, heldsway. But from 1801 to 1803, Thomas Young carried out well-designed diffractionexperiments that put the wave theory of light back in the race. Although Young’s ex-periments (especially the double-slit experiment published in 1807) dealt a body blowto the corpuscular theory, light waves were still considered to be vibratory motion in thelongitudinal direction, along the direction of propagation. As such, they could not ac-count for polarisation. In 1821, the French engineer Augustin-Jean Fresnel (1788–1827)showed that polarisation was consistent with the wave picture if the periodic vibrationwas transverse to the direction of propagation. This finding removed the principal re-maining objection to the wave model of light. The dramatic Arago spot demonstration,illustrated in Figure 1.6, in which a bright central spot can be observed in the geometricshadow of an opaque circular screen, laid to rest the corpuscular model forever (or so itseemed at the time).

At the same time that Young and Fresnel were correctly characterising the wave na-ture of light, Michael Faraday (1791–1867) was carrying out experiments in electricityand magnetism. Faraday pictured magnetic influences acting on bodies not in directcontact as lines of force. These lines, originating and terminating in closed loops, were thebeginnings of force fields acting on bodies through space with no actual physical contact.Together with Newton’s theory of gravitation, these ideas elevated action-at-a-distanceto serious consideration. The next step was the great unification of electric, magnetic,and optical phenomena effected by James Clerk Maxwell (1831–1879). In 1865, Max-well published A Dynamical Theory of the Electromagnetic Field in which he set forth theproposition that light was in fact a transverse electromagnetic wave. In modern form, thefour Maxwell equations and the Lorentz force law constitute a unified classical theory ofelectricity, magnetism, and light.

While Young, Fresnel, and Maxwell were putting light, electricity, and magnetismon a firm foundation, an understanding of ponderable matter and chemical interaction


–80

0.2

0.4

0.6

Rel

ativ

e in

tens

ity/a

.U.

0.8

1

1.2

1.4

–6 –4 –2 0r/mm

Poisson Spot

Screen with shadowof circular object.

Screen which casts acircular shadow

Point light source

Fresnel’s entry in the French Academy Competition of1818 to explain the Young double-slit experiment.

2 4 6 8

Figure 1.6 The celebrated Arago or Poisson spot experiment diagrammed on the left with the simulatedinterference measurement and intensity trace through the centre on the right. All figures, CreativeCommons by-sa 3.0, Thomas Reisinger.

was also advancing rapidly. Building on Boyle’s experiments with gases, Joseph Priestly(1733–1804) conducted extensive experiments on ‘airs’ and identified nitric (NO) andnitrous (N2O) oxides, and oxygen (O2), which he called ‘dephlogisticated air’. Thisterm refers to ‘phlogiston’, a substance that was thought to be contained within matterand expelled during combustion. Although now discredited by modern understandingof oxidation, Priestly used the principle of phlogiston to rationalise his observations.In France, Joseph Louis Gay-Lussac (1778–1850) annunciated another physical gaslaw, complementary to Boyle’s law. Gay-Lussac found that for a given quantity of gas,the volume was directly proportional to the temperature at constant pressure. Back inEngland, John Dalton (1766–1844) ascertained that elements combine in simple num-ber ratios, and he began to determine atomic masses. Dalton also determined that thepressure of a mixture of gases was equal to the sum of the pressures of the individualconstituents. However, the major unifying advance in understanding chemical reactiv-ity was carried out by Antoine Lavoisier (1743–1794), considered the father of modernchemistry. Lavoisier was the first to clearly annunciate that between reactants and prod-ucts of a chemical reaction, mass was conserved. The recognition of mass conservationas a fundamental principle distinguished light, now recognised as an electromagneticfield wave, from ponderable matter. Finally, Dmitri Mendeleev (1834–1907) unified thegrowing list of elements into a rational order with the periodic table in 1869. Figure 1.7shows Mendeleev’s early periodic chart.

One scientific revolution spawns another 9

Dmitri Mendeleev

Figure 1.7 Dmitri Mendeleev and his early periodic chart of the elements. Figures in the publicdomain.

As the nineteenth century drew to a close, there were grounds for great satisfac-tion. Although the inner structure of the atomic elements was not known, their chemicalidentity and relationships to each other were firmly established. The periodic table notonly rationalised known elements but predicted those not yet discovered, leading OliverLodge to write in 1888: ‘The whole subject of electrical radiation seems to be work-ing itself out splendidly’. There was only the somewhat perplexing null result of theMichelson–Morley experiment in 1887 that failed to measure the velocity of the ‘lu-miniferous aether’ through which light was supposed to travel, and the failure of theequipartition of energy principle to account for the spectral distribution of black-bodyradiation. But these were considered minor blemishes on a near perfect masterwork ofhuman understanding.

1.5 One scientific revolution spawns another

The null result of the Michelson–Morley experiment and its subsequent refinementswere finally explained by the theory of special relativity proposed in 1905 by AlbertEinstein (1879–1955). The principal elements of the Michelson–Morley experimentsare depicted in Figure 1.8. Special relativity showed that the speed of light in a vacuumwas not just a property of the electromagnetic wave but a universal constant independentof the inertial reference frame in which it is measured. It follows that any experimentattempting to determine the luminiferous aether wind, with respect to the earth’s motion,must yield a null result. Furthermore, in a subsequent paper published later in the same


a

uc

m b g

dk

clo

Sun

Luminiferous Ether

Earth(fall)

Earth(spring)

Figure 1.8 The Michelson–Morley experiment. The diagram on the left shows how the speed of light,travelling through a ‘luminiferous aether’, should show a Doppler shift with respect to the motion of theearth in its orbit around the sun. As the earth moves toward (away) from the direction of propagation oflight from a distant source, the shift should be blue (red). The schematic on the right shows Michelson’sinterferometer setup that was designed with sufficient resolution such that the shift in the interferencepattern would be measurable. Left figure, © cc by-sa 3.0, Cronholm144. Right top figure, publicdomain. Right bottom figure, Creative Commons by-sa 2.5, Alain Le Rille.

year, Einstein showed that special relativity implied the equivalence of matter and energythrough the famous formula, m = E/c2.

The ‘new philosophy’ originating with Descartes and culminating in the achievementsof Maxwell andMendeleev appeared to establish a fundamental distinction between lightand matter. Within a few decades of their master works, however, special relativity an-nounced a fundamental equivalence between them. Furthermore, in a subsequent paperof the same year, Einstein proposed the quantisation of radiation that re-establishedsome of the old Newtonian corpuscular properties of light. Energy quanta in the formof E = hν, where ν is the frequency of light and h is Planck’s constant, also removed the‘ultraviolet catastrophe’ from the Rayleigh-Jeans formulation of black-body radiation.Finally, the quantum mechanics established by Max Born, Erwin Schrödinger, WernerHeisenberg, and Paul Dirac in the first decades of the twentieth century explained theperiodic table of the chemical elements in terms of the inner structure of atoms. Quan-tum mechanics in its non-relativistic form is quite adequate for an understanding ofmatter outside the atomic nucleus. Together with Dirac’s initial formulation of a quant-ised version of electromagnetism, the second revolution in physics was almost complete.In 1916, Einstein was able to generalise his theory of relativity and apply it to gravity.

Summary 11

The result was a new geometric conception of ‘space-time’ relevant to cosmological en-ergy and length scales. Of the four known classes of forces—gravity, electromagnetism,the strong, and the weak, nuclear forces—only two have been unified into a consistenttheoretical framework: electromagnetism and the weak force. To date, a quantum the-ory of gravity does not exist and current efforts to unify the force classes into a ‘theoryof everything’ using string theory or M-theory continue. However, for the purposes ofthis book, classical electrodynamics, essentially as formulated by the Maxwell’s equationsand the Lorentz force law, together with non-relativistic quantum mechanics, is perfectlyadequate to describe light–matter interactions.

1.6 Summary

In this chapter we have briefly examined notions of light and matter from antiquityto the present day, and the major developments are illustrated in Figure 1.9. The first

Greek Atomists-5th century BC

Descartes-17th century ADmatter and motion

Huygens-light is awave

Young-interference

Fresnel-diffraction

Fizeau, Foucault-speed of light

Maxwell-electromagneticfield

m = E/c2

E = hv

Newton-matteris mass

Lavoisier-massconserved inchemical reaction

Mendeleev-periodic table

Planck-quantum energy

Einstein-special relativity

mass is proportional tolight frequency

lightmatter

Empedocles-4 elements

Euclid-optics

m =c2h v

Figure 1.9 Major developments in the history of light and matter.


ideas were embedded in speculative philosophical ‘theories of everything’ without muchthought to testing them through experimentation. Evidence of a more modern, scientificapproach was found in Alhazen’s Book of Optics, a treatise very influential in the Westduring the European Renaissance. Galileo, Grimaldi, Hooke, Huygens and Newton es-tablished major milestones in the seventeenth century on the route to understandingthe nature of light and its interaction with matter. Newton’s conception of the corpus-cular character of light held sway for most of the eighteenth century until the theorycame under serious attack at the beginning of the nineteenth century from Fresnel andYoung, who demonstrated convincing experimental evidence that light was really a wave.The proposition that light was a transverse wave removed the last defensive rampart ofthe Newtonian corpuscular school, and James Clerk Maxwell produced the crowningachievement with the Theory of the Electromagnetic Field in 1865. In the meantime, the-ories of matter evolved from a philosophical, speculative atomism of antiquity, throughthe Cartesian ‘plenum’ to Boyle’s and Priestly’s experiments with gases that lent cred-ibility to integer combining numbers (volumes). Dalton produced evidence for chemicalcombinations of whole numbers and began to determine atomic masses, while the uni-fying achievement of Mendeleev’s periodic table of the elements lent predictive powerto the atomic theory of matter. Black-body radiation and the Michelson–Morley experi-ment showed that the triumphalism of the late nineteenth century was premature, andthat quantal atomic structure, special relativity, and the quantisation of the radiationfields were required to explain not only those two experiments but also the entire bodyof atomic and molecular spectroscopy. The bifurcation of the antique world into matterand light now seems to be returning to a more unified view of the equivalence of lightand matter.

1.7 Further reading

1. E. T. Whittaker, A History of the Theories of Aether and Electricity: From the Age ofDescartes to the Close of the Nineteenth Century, BiblioLIfe, Charleston, South Carolina(1910).

2. F. Wilczek, The Lightness of Being, Basic Books, New York (2008).

3. S. Hawking and L. Mlodinow, The Grand Design, Bantam Books, New York (2010).

4. J. Gribbin, The Scientists, Random House, New York (2004).

5. D. C. Lindberg, The Beginnings of Western Science, University of Chicago Press,Chicago (2007).

6. C. A. Pickover, Archimedes to Hawking, Oxford University Press, Oxford (2008).

2

Elements of Classical Electrodynamics

2.1 Introduction

Before we can discuss the interaction of light and matter we have to establish a workingvocabulary for the vector fields of classical electrodynamics and the equations that relateto them. These equations, Maxwell’s equations, together with the Lorentz force law (orsome other force law), serve as the postulates of that theory. A class of solutions tothese equations describes the propagation of plane electromagnetic waves, and we willexplore this propagation in vacuum, in dielectrics, and in good conductors. We will alsostudy polarisation fields in bulk matter, the relation between the macroscopic polarisationfield and the microscopic atomic polarisability, and finally develop further the centralrole played by the radiating dipole in the construction of antennas and antenna arrays.Later, in Chapter 6, we will also describe how standing waves are established in a cavityand how to calculate the energy density by counting modes. The energy density is thenapplied to the problem of classical black-body radiation theory, and we will describe howit leads to a fundamental contradiction between the predictions of classical theory andexperimental measurement.

2.2 Relations among classical field quantities

Since virtually all students now learn electricity and magnetism with the SI (SystèmeInternational) or rationalized MKS (metres, kilograms, seconds) family of units, we ad-opt it here. A discussion of units and fundamental quantities in electromagnetism ispresented in Appendix B. There we will see that although SI is now the most commonchoice, other unit systems have their advantages, and in any case, all expressions can bereadily transposed from one system to another.

The present choice of SI means that we write Coulomb’s force law between twoelectric point charges q, q′ separated by a distance r as

F =1

4πε0

(qq′

r3

)r (2.1)


14 Elements of Classical Electrodynamics

and Ampère’s force law (force per unit length) of magnetic induction between twoinfinitely long wires carrying electric currents I , I ′, separated by a distance r as

d |F|d l

=μ0

2π

(II ′

r

)(2.2)

where ε0 and μ0 are called the permittivity of free space and the permeability of free space,respectively. In this unit system, the permeability of free space is defined as

μ0

4π≡ 10–7 (2.3)

and the numerical value of the permittivity of free space is fixed by the condition that

1ε0μ0

= c2 (2.4)

where c is the speed of light in vacuum. Therefore we must have

14πε0

= 10–7c2 (2.5)

The time-independent vector fields issuing from these force expressions are the electricfield, E, and the magnetic induction field, B. The E-field is the Coulomb force per unitcharge,

E =1

4πε0

qr3r (2.6)

and the magnetic induction B-field is given by the Biot–Savart law,

B =μ0

4π

∫I× rr3

dl =μ0

4πI∫dl× rr3

(2.7)

where I is the current running in a wire and dl is an element of the wire length. TheLorentz force law succinctly summarises the effect of the E- and B-fields on a chargedparticle moving with velocity v,

F = q(E + v×B) (2.8)

In addition to these two fields, the displacement fieldD and magnetic fieldH are neededto describe the modification of force fields in dielectric or conductive matter. In linear,

Classical fields in matter 15

isotropic materials, these two additional fields are linked to E and B by the ‘constitutiverelations’

D = εE (2.9)

H =1μB (2.10)

where ε and μ are the permittivity and permeability of the material, respectively. Thesematerial parameters are related to those of free space by

ε = ε0εr (2.11)

μ = μ0μr (2.12)

where εr and μr are the relative (and unitless) permittivity and permeability, respect-ively. The relative permittivity εr is often called the dielectric constant, but the relativepermeability rarely finds application and does not have a common alternative name.Unfortunately, many authors use ε for the dielectric constant, so there is danger of con-fusion between permittivity (with MKS units of C2/N · m2) and the unitless dielectricconstant. However, one can always discern from the context what is meant by the symbolε. There is also some difference of opinion among authors as to the most appropriatenomenclature for the B-field and H-field. Here we follow conventional usage and call theB-field the ‘magnetic induction field’, and the H-field, the ‘magnetic field’.

2.3 Classical fields in matter

In free space, E,B,D, and H are related by

D = ε0E (2.13)

B = μ0H (2.14)

When these force fields act on a material medium, however, the D- and H-fields take onadded terms. Matter consists of positively charged core nuclei surrounded by distribu-tions of negatively charged electrons. If the core nuclei are arranged according to somesymmetric spatial extension the material is crystalline, and if not, the material may be aglassy solid, a liquid, or a gas. The electric charge distribution may be bound to the nu-clei or delocalised throughout the crystal structure. In any case, if E and B are present,the Lorentz forces acting on the electric charge distribution will clearly modify it. Thesemodifications can be characterised by the introduction of two new fields: the polarisation


P and the magnetisation M. In the presence of matter, the displacement and magneticfields now are expressed as

D = ε0E + P (2.15)

B = μ0H +M (2.16)

For E- and B-fields that are not too strong, the P- and M-fields themselves areproportional to E and H:

P = ε0χeE (2.17)

M = μ0χmH (2.18)

where χe, χm are electric and magnetic susceptibility. Equations 2.15 and 2.16 thenbecome

D = ε0(1 + χe)E (2.19)

B = μ0(1 + χm)H (2.20)

and from Equations 2.9 and 2.10 we see that the relative permittivity and permeabilitycan be written in terms of the corresponding susceptibilities as

εr = 1 + χe (2.21)

μr = 1 + χm (2.22)

The relative permittivity and permeability are unitless but may be complex; theimaginary parts reflecting absorptive loss:

εr = ε′ + iε′′ = 1 + χ ′e + iχ′′e (2.23)

μr = μ′ + iμ′′ = 1 + χ ′m + iχ ′′m (2.24)

2.4 Maxwell’s equations

In the foregoing, we have presented the fields of classical electrodynamics, D,E,B,H,and the fields induced in materials, P and M, which represent the response of matter

Maxwell’s equations 17

to electric and magnetic forces. The equations governing the spatial and temporalbehaviour of these fields are Maxwell’s equations,

∇ ·D = ρfree (2.25)

∇ ·B = 0 (2.26)

∇ × E = –∂B∂t

(2.27)

∇ ×H = Jfree +∂Ddt

(2.28)

The first two equations make statements about field sources. The first states that thesource of the displacement field D is the ‘free’ electric charge density, ρfree. In fact, thetotal charge density is composed of two terms: the free charge density and the ‘bound’charge, ρbound:

ρ = ρfree + ρbound (2.29)

The bound charge density is defined as the negative divergence of the polarisation field.In Chapter 10, Section 10.4.3, we will see what motivates that definition:

ρbound = –∇ · P (2.30)

so that from Equations 2.15, 2.25, and 2.30 we see that

∇ ·E =ρ

ε0(2.31)

The second source equation, Equation 2.26, states that the magnetic induction field Bdoes not originate from a magnetic charge density, ρm. In fact, Equation 2.26 impliesthat magnetic source ‘charges’ do not exist: magnetic monopoles have never been foundin nature (although Dirac’s quantum electrodynamics implies that they must exist some-where or have existed at some time). The second pair of equations, termed Faraday’s lawand the Maxwell–Ampère law, respectively, describe the spatial and temporal behaviourof the fields. Equation 2.28, Ampère’s law, introduces another field, the charge currentdensity, Jfree. Just as Equation 2.29 expresses the total charge density as the sum of thefree and bound charge densities, so the total current density J is composed of the sumof free and bound current densities:

J = Jfree + Jbound (2.32)

The bound current density is defined by two contributions: one from the polarisationfield and the other from the magnetisation field:

Jbound =∂P∂t

+1μ0

∇ ×M (2.33)


and from Equation 2.28

Jfree = –∂D∂t

+ ∇ ×H (2.34)

We see how D and H play analogous roles for Jfree as P and M play for Jbound. Theterms ∂P/∂t and ∂D/∂t are called the polarisation current and the displacement current,respectively.

2.4.1 Charge-current continuity

If we take the divergence of both sides of the Maxwell–Ampère law (Equation 2.28) anduse Equation 2.25, we find

∇ · Jfree + ∂ρfree

∂t= 0 (2.35)

This equation is called the charge-current continuity equation and states that the freecurrent leaving or entering a closed surface must be equal to the time rate of free chargein that volume. The continuity condition is important because it implies that currentand charge sources of fields D and H cannot be independently specified. They areconstrained by charge-current conservation.

2.5 Static fields, potentials, and energy

2.5.1 Electric field energy

The E-field resulting from a single point charge, Equation 2.6, can be written as

E =1

4πε0

qr2r (2.36)

where r is the distance from the charge and the radial unit vector r = r/r. Written inthis way, the familiar 1/r2 radial fall-off of the field is made explicit. Any vector field isspecified by the divergence and the curl of the field, and in the case of Equation 2.36,the divergence is

∇ · E =1ε0ρ(r) (2.37)

where ρ(r) is the charge density at the position r.It can be easily shown by the application of Stokes’ theorem (see Appendix C for

a discussion of the differential and integral operations common in vector fields) toEquation 2.36 that the curl of E is always null:

∇ × E = 0 (2.38)

Static fields, potentials, and energy 19

By the principal of vector field superposition, this property is true not just for the E-fieldof a single point charge but for any spatial distribution of point charges. The E-field issaid to be irrotational and therefore can be set equal to the gradient of a scalar function,say,

E = –∇V (2.39)

This characterisation of the E-field as the gradient of a scalar potential V is justified bythe vector calculus identity that the curl of the gradient of any scalar function is null:

∇ × E = –∇ × ∇V = 0 (2.40)

The negative sign on the right-hand side of Equation 2.39 is a convention. A unit analysisshows that V has units of energy per charge, and the work done to bring n charges frominfinity to a given spatial configuration is

Eelec =12

n∑i=1

qiV (ri) (2.41)

The factor of 1/2 on the right-hand side of Equation 2.41 avoids double counting ofmutual pairwise interactions among charges. This expression can be generalised to asmooth charge density distribution ρ(r),

Eelec =12

∫ρV dτ (2.42)

and taking into account Equations 2.37, and 2.39, the energy required to assemble thecharge distribution can be expressed in terms of the resulting E-field,

Eelec =ε0

2

∫E2 dτ (2.43)

where the integration is over all space. Equation 2.43 can simply be interpreted as theenergy of the E-field.

2.5.2 Magnetic field energy

We know from Equation 2.26 that the divergence of the B-field is null,

∇ ·B = 0 (2.44)

but we find from the Biot–Savart law describing the magnetic field issuing from currentflowing in a wire, Equation 2.7, that the curl of the B-field is given by

∇ ×B = μ0J (2.45)


where J is the distribution of the source current densities giving rise toB. Just as we intro-duced the scalar potential V (r) to characterise the curl-less E-field, so we can introducea vector potential A to characterise the divergence-less B-field,

B = ∇ ×A (2.46)

and setting the divergence of the vector potential to zero, we find the standard expressionfor A is

A(r) =μ0

4π

∫J(r′)r

dτ ′ (2.47)

where r = |r–r′|, the distance between the point rwhereA is evaluated and r′ the positionof the current density J. Just as it is possible to write the energy of some distribution ofelectric charge as an integral over the product of the charge density distribution ρ(r)and the scalar potential V (r) (Equation 2.42), we can write the magnetic energy as theintegral over a scalar product of the current density distribution J(r) and the vectorpotential A(r).

Emag =12

∫J ·A dτ (2.48)

By recognising that J is proportional to the curl of B from Equation 2.46, substitutinginto Equation 2.48, and integrating by parts, we can write the magnetic energy entirelyin terms of the B-field in an expression analogous to Equation 2.43

Emag =1

2μ0

∫B2 dτ (2.49)

where the integral is taken over all space.

2.5.3 Poynting’s theorem

From Equations 2.43 and 2.49 it appears plausible that the energy of an electromagneticfield is the sum of the energy of the constituent parts,

Eem =12

∫ (ε0E2 +

1μ0B2)dτ (2.50)

and this result can, in fact, be obtained directly from one of the basic postulates ofclassical electrodynamics, the Lorentz force law,

F = q(E + v×B) (2.51)

Static fields, potentials, and energy 21

The force F acting on charged particles set in motion with velocity v does work on theparticles at the rate

dEmech

dt= qE · v (2.52)

where Emech is the mechanical energy of the particle system. Of course, the B-field,always acting at right angles to the direction of motion, can do no work on the particles.But q =

∫ρdτ and v = Jfree/ρ, so

dEmech

dt=∫

E · Jfree dτ (2.53)

Now we can use the Maxwell–Ampère law, Equation 2.28, to eliminate Jfree in Equa-tion 2.53. The dot product in the integrand becomes

E · Jfree = E · (∇ ×H) – E · ∂D∂t

(2.54)

The next step is to invoke a vector field identity and the Faraday law, Equation 2.27, torewrite the first term on the right of Equation 2.54. The vector field identity is

∇ · (E×H) = H · (∇ × E) – E · (∇ ×H) (2.55)

and Faraday’s law is

∇ × E = –∂B∂t

so

E · (∇ ×H) = –H · ∂B∂t

– ∇ · (E×H) (2.56)

and

E · Jfree = –H · ∂B∂t

– E · ∂D∂t

– ∇ · (E×H) (2.57)

Now we substitute the E · Jfree expression back into the integrand in Equation 2.53, anduse Stokes’ theorem to convert the volume integral of the ∇ · (E ·H) term to a surfaceintegral with integrand E ·H. The resulting expression for the time rate of work done bythe electromagnetic field on the charged particles is

dEmech

dt= –

∫ (H · ∂B

∂t+ E · ∂D

∂t

)dτ –

∮σ

(E×H) · da (2.58)


where the integral of the second term on the right is taken over the surface σ boundingthe volume integral in the first term. The two terms in the volume integral are identifiedwith the electric and magnetic parts of the electromagnetic field energy. In a free-spacevolume, or in any medium with negligible polarisation, the decrease in the field energy,as work is being done on the system of particles, can be written as

–dEem

dt= –

ddt

∫12

(1μ0B2 + ε0E2

)dτ (2.59)

confirming that the total electromagnetic field energy can be identified as the sum ofthe separate static magnetic and electric terms as in Equation 2.50. We can now writeEquation 2.58 as an energy conservation expression,

dEmech

dt+dEem

dt= –

∮E×H · da (2.60)

where the term on the right is interpreted as an integral over the energy flux (energyper unit area per unit time) flowing across the bounding surface. This cross-productexpression is termed the Poynting vector, S:

S = (E×H) (2.61)

Notice that energy increasing within the volume is equal to a negatively signed energyflux into the volume across the boundary surface. The negative sign on the Poyntingvector is simply a consequence of the sign convention on the vector surface differentialda, positive pointing outwards.

2.6 Three examples of problem solving in electrostatics

The following three examples illustrate many of the useful analytical tools needed to solveproblems in electrostatics. They are graded in increasing difficulty, and understandingevery line is not essential in a first reading (especially the third example), but the use ofcylindrical and spherical coordinates for the appropriately symmetric problem, and theconsequent separation of variables, needs to be thoroughly understood by any seriousstudent of light–matter interaction. The same strategy will apply when we consider thestructure of the hydrogen atom in Chapter 12. Gauss’s law plays a pivotal role in manyof the applications we will study in the following chapters. We write out in detail herethe solutions to three problems; the results of which will be useful in subsequent topics.The integral form of Gauss’s law is also developed at the end of this section.

Three examples of problem solving in electrostatics 23

Example 1

Estimate the electron oscillation frequency in a hydrogen atom using the differential form ofGauss’s law.

Solution 1

To solve this problem, let us consider the following model for the hydrogen atom. We assumethat the electron is represented by a cloud of charge density ρ, symmetrically distributed in asphere of radius R centred at the origin x = 0 where a positively charged proton is located:

ρ =eV

=e

4/3πR2(2.62)

The force of attraction between the proton and the electron charge distribution is given byCoulomb’s law:

F = –e2

4πε0r2r (2.63)

Under the action of a constant external E-field, the cloud will be displaced from its equilibriumposition. Let us call x the displacement distance between charge centres at which a new balanceof forces will occur. Figure 2.1 illustrates this displacement. We seek to calculate the restoringforce on the electron charge density using Gauss’s law. This restoring force will be exerted bythe charge fraction δq located at the surface of the sphere with radius x. The fraction of chargeoutside the volume Vx = 4/3πx3 produces no force on the proton. Taking ρ as the electroncharge density, we have

δq = (ρ)(Vx) = –e( xR

)3. (2.64)

R

E

X

+–

Figure 2.1 Displacement of the electron charge cloud under theinfluence of a constant external E-field.

continued


Example 1 continued

The magnitude of the effective Coulomb force between the proton and the displaced electroncharge cloud is now

|F| = eδq

4πε0x2= –

e2

4πε0x2x3

R3= –Kx (2.65)

which shows that small displacements result in a linear restoring force with the forceconstant K :

K =e2

4πε0R3(2.66)

Remembering that the frequency of oscillation is given by

ω =

√Km

=

√e2

4πε0R3me(2.67)

and taking R = 0.529–10 m as the radius of the hydrogen atom in the ground state, andme = 9.109 × 10–31 kg, the rest mass of the electron, we calculate the charge frequency ofoscillation ωe = 4.13× 1016 s–1.

Example 2

Calculate the E-field of a three-dimensional static electric dipole from the potential.

Solution 2

In Section 2.8 we will study the oscillating dipole and will emphasize the time dependenceof the field and radiation. Here we study the static dipole and illustrate the usefulness ofcalculating the E-field by first writing down the scalar potential, then taking its divergence.

A dipole is a distribution of two equal but opposite charges, separated by a distance a asshown in Figure 2.2. The dipole is a vector, which by convention points from the negative tothe positive charge. In Figure 2.2 the dipole p is oriented along z, p = qaεz. The calculationof the E-field is carried out through the electric potential using E = –∇V (x, y, z). Since thedipole is oriented along the z-axis and has cylindrical symmetry around this axis, the dipolefield is invariant with rotation about the azimuthal angle ϕ. From Figure 2.2, at point r we havefor the potential,

V (x, y, z) =q

4πε0

(1r1

–1r2

)(2.68)


Example 2 continued

z

A

r1

r2

y

x

p

r+q

−qϕ

-a/2

a/2

θ

Figure 2.2 Electric dipole in a system of polar andcartesian coordinates.

Since we know from the law of cosines,

r21 = r2 +( a2

)2– a r cos θ and r22 = r2 +

( a2

)2+ a r cos θ (2.69)

Equation 2.68 takes the form

V (r, θ) =q

4πε0

⎡⎢⎣ 1(

r2 +( a2)2 – ar cos θ

)1/2 –1(

r2 +( a2)2 + ar cos θ

)1/2⎤⎥⎦ (2.70)

Taking advantage of the dipole cylindrical symmetry, we calculate the electric field by applyingthe gradient operation to the potential in polar coordinates,

E(r, θ) = Er εr + Eθ εθ (2.71)

with

Er = –∂V∂r

and Eθ = –1r∂V∂θ

(2.72)

continued


Example 2 continued

Carrying out the derivative operations on Equation 2.70 we find

Er =q

4πε0

⎡⎢⎣ r – a

2 cos θ(r2 +

( a2)2 – ar cos θ

)3/2 –r + a

2 cos θ(r2 +

( a2)2 + ar cos θ

)3/2⎤⎥⎦ (2.73)

and

Eθ =q

4πε0

⎡⎢⎣ a

2 sin θ(r2 +

( a2)2 – ar cos θ

)3/2 +a2 sin θ(

r2 +( a2)2 + ar cos θ

)3/2⎤⎥⎦ (2.74)

In many practical problems, the dipole potential and field are of interest when r� a: the regionof space termed the far field. Under these conditions we can write[

r2 +( a2

)2 ± a r cos θ]n = r2n[1 +

( a2r

)2 ± (ar

)cos θ

]n� r2n

[1 + n

[( a2r

)2 ± (ar

)cos θ

]](2.75)

and therefore write the dipole potential Equation 2.68 in the far field as

V (r, θ) =q

4πε0

1r

{[1 –

12

[( a2r

)2–(ar

)cos θ

]]–[1 –

12

[( a2r

)2+(ar

)cos θ

]]}(2.76)

which, after simplification, yields

V (r, θ) � qa cos θ

4πε0r2(2.77)

and the far-field dipole E-field

E(r, θ) =2qa cos θ

4πε0r3εr +

qa sin θ

4πε0r3εθ (2.78)

We can write the dipole potential and E-field, Equations 2.77 and 2.78, in vector form as

V (r, θ) =p · εr4πε0r2

(2.79)

and

E(r, θ) =(2p · εr)εr – (p · εθ )εθ

4πε0r3(2.80)

where εr, εθ are unit vectors in polar coordinates. As discussed at some length in AppendixD,the dipole vector E-field can be expressed either in polar or cartesian coordinates. The unit


Example 2 continued

vectors of the E-field in polar coordinates are εr, εθ , εϕ , and the dipole p in Figure 2.3 can bewritten in terms of the polar components as

p =(p · εr

)εr +

(p · εθ

)εθ +

(p · εϕ

)εϕ (2.81)

The relations between the unit vectors in polar coordinates and cartesian coordinates aregiven by

εr = εx sin θ cosϕ + εy sin θ sinϕ + εz cos θ (2.82)

εθ = εx cos θ cosϕ + εy cos θ sinϕ – εz sin θ (2.83)

εϕ = –εx sin ϕ + εy cosϕ (2.84)

By multiplying the first of these relations by cos θ and the second by – sin θ we can easilyshow that

cos θ εr – sin θ εθ = εz (2.85)

In Figure 2.3 we see that in an r, θ plane, carrying out the dot products in Equation 2.81 withthe help of Equations 2.82, 2.83, and 2.84 results in

p = p(cos θ εr – sin θ εθ ) (2.86)

p sin θ

p cos θ

z

p

θ

Figure 2.3 Relationbetween cartesian and polarcoordinates for dipoleoriented along z-axis.

continued


Example 2 continued

and therefore

(p · εθ )εθ = –p sin θ εθ = p – (p cos θ)εr = p – (p · εr)εr (2.87)

Substituting this last expression into Equation 2.78 results in

E(r, θ) =(3p · εr)εr – p

4πε0r3(2.88)

Example 3

Analyse the polarisation of a dielectric sphere of radius R immersed in a uniform electric fieldwith permittivity ε.

Solution 3

Consider a dielectric sphere placed in a uniform electric field E as shown in Figure 2.4. Weanalyse the problem in polar coordinates. The potential along the z direction is given by

V = –E · z = –E0r cos θ (2.89)

E0

r

r0

y

z

P

++

+

+

++

+––

––

––

–

θ

Figure 2.4 Dielectric sphere immersed in a uniform electric fieldoriented along the z direction.

Our first goal is to determine the potential within, and external to, the sphere. To find thepotential we solve Laplace’s equation,

∇2V = 0 (2.90)


Example 3 continued

in both regions and then join the solutions at the boundary. We show in Appendix D, Equa-tionD.37 how to express the Laplacian operator in polar coordinates. An equivalent, slightlydifferent form is

∇ ·∇V =1r∂2

∂r2(rV ) +

1

r2 sin θ· ∂∂θ

(sin θ

∂V∂θ

)+

1

r2 sin2 θ

∂2V

∂ϕ2(2.91)

The problem of the dielectric sphere subject to a constant electric field along the z-axis isobviously cylindrically symmetric. Before attacking this problem directly, we examine the formof the solutions to Laplace’s equation for spherical symmetry. Assuming for the moment aspherically symmetric potential V , we write V as a product of a radial function R(r)/r, a polarangular function �(θ), and an azimuthal angular function (ϕ):

V (r, θ) =R(r)r

�(θ)(ϕ) (2.92)

Now we substitute this form back into Equation 2.91 and arrange all terms with r and θ de-pendence on the left-hand side, and all terms with ϕ dependence on the right-hand side. Aftersome algebra, the result is

r2 sin2 θR(r)

∂2R(r)

∂r2+

sin θ�(θ)

[cos θ

∂�(θ)∂θ

+ sin θ∂2�(θ)

∂�2(θ)

]= –

1(ϕ)

∂2

∂ϕ2(2.93)

Since r, θ , ϕ are independent variables, the only way that Equation 2.93 can be true is for eachside to be equal to a constant, called the ‘separation constant’. We can choose the form ofthis constant to be anything we want, so we choose it to be m2. From the right-hand side ofEquation 2.93 we then have

1(ϕ)

∂2

∂ϕ2= –m2 (2.94)

and we can see by simple inspection, and can verify by substitution, that the solutions are

(ϕ) = e–imϕ (2.95)

In order for to be a single-valued function in ϕ (modulo 2π), m must be an integer, m =0,±1,±2, . . .. After division by sin2 θ , the left-hand side of Equation 2.93, also equal to m2,can be rearranged to

r2

R(r)∂2R

∂r2+

1sin θ�(θ)

[cos θ

∂�(θ)∂θ

+ sin θ∂2�(θ)

∂θ2

]–

m2

sin2 θ= 0 (2.96)

The first term in Equation 2.96 depends only on r and the next two only on θ . Therefore,they can also be separated and set equal to a separation constant, the form of which we chooseto be l(l + 1):

continued


Example 3 continued

r2

R(r)d2R

dr2= l(l + 1) (2.97)

–

{1

(sin θ)�(θ)

[cos θ

d�(θ)dθ

+ sin θd2�(θ)

dθ2

]–

m2

sin2 θ

}= l(l + 1) (2.98)

Equation 2.98 can be made more concise by recognising that the term in square brackets is anexpanded differential of a product of functions. Multiplying both sides by � we have

1sin θ

ddθ

(sin θ

d�dθ

)–

m2

sin2 θ� = –l(l + 1)� (2.99)

The radial and angular differential equations can finally be rearranged to

d2R

dr2–l(l + 1)

r2= 0 (2.100)

d2�

dθ2+ cot θ

d�dθ

+

[l(l + 1) –

m2

sin2 θ

]� = 0 (2.101)

We have already observed that m must be zero or an integer in order to maintain as asingle-valued function. In this field-matching problem of a dielectric sphere immersed in aconstant electric field, oriented along z, the E-field will show no ϕ dependence, and we canset m = 0. The resulting expression is called Legendre’s equation:

d2�

dθ2+ cot θ

d�dθ

+ [l(l + 1)]� = 0 (2.102)

Often, the independent variable of the Legendre equation is taken to be x = cos θ rather thanθ itself. Then Equation 2.102 takes the form

ddx

[(1 – x2

) d�dx

]+ l(l + 1)� = 0 (2.103)

The physically admissible solutions to Equation 2.103 are a family of polynomials in cos θcalled the Legendre polynomials. They are labelled by the index l which must only assumethe positive integer values l = 0, 1, 2, . . .. The first few Legendre polynomials are listed inTable 2.1 The Rodrigues’ formula is a concise expression of any desired member l of theLegendre polynomials

Pl(x) =1

2l l!

dl

dxl

(x2 – 1

)l(2.104)

The solution to the separated radial equation, Equation 2.100, is

R(r) = Arl + Br–(l–1) (2.105)


Example 3 continued

where A and B are, as yet, undetermined constants. The fact that Equation 2.105 really is asolution can be verified by direct substitution.

Table 2.1 Legendre polynomials

Index l Function Polynomial

0 P0(x) 11 P1(x) x

2 P2(x) 1/2(3x2 – 1

)3 P3(x) 1/2

(5x3 – 3x

)4 P4(x) 1/8

(35x4 – 30x2 + 3

)

Nowwe are in a position to write the general solution, Equation 2.92, toMaxwell’s equationgoverning the response of a dielectric sphere immersed in a constant electric field:

V (r, θ)r

= U(r, θ ,ϕ) =∞∑l=0

[Arl + Br–(l+1)

]Pl(cos θ)e

–imϕ (2.106)

Because the problem has cylindrical symmetry, and the solution is invariant about the z-axisof the applied E-field, the m = 0 solution of (ϕ) = e–imϕ is the appropriate choice, and(ϕ) = 1. Furthermore, we must fit the solution to the physical boundary conditions at theorigin, at infinity, and at the interface on the surface of the sphere. Since the potential mustremain finite as r→ 0, it is clear that we should choose B = 0 for r inside r0, the radius of thesphere. In this region

Uin(r, θ) =∞∑l=0

AlrlPl(cos θ) (2.107)

Outside the sphere, in the limit r→∞ we see from Equation 2.89 that the asymptotic solutionmust take on the form –E0 cos θ . In order to fit this result, A1 = –E0, with all other Al = 0.The solution outside the sphere must therefore have the form

Uout(r, θ) = –E0r cos θ +∞∑l=0

Blr–(l+1)Pl(cos θ) (2.108)

In order to determine the remaining parameters Al and Bl we apply continuity conditions atthe sphere surface. The solutions must match there, and in order to match smoothly, the firstderivatives must match as well. Therefore, we have at r = r0

∞∑l=0

Alrl0Pl(cos θ) = –E0r0 cos θ +

∞∑l=0

Blr–(l+1)Pl(cos θ) (2.109)

continued


Example 3 continued

for the first continuity condition. For the second condition, we remember that

Er = –∂U(r, θ)∂θ

(2.110)

and that components of the D-field normal to the boundary must be continuous. Then wehave for the second continuity condition

– εin

∞∑l=0

Allr(l–1)0 Pl(cos θ) = –εout

⎡⎣–E0r0 cos θ + ∞∑

l=0

Blr–(l+1)0 Pl(cos θ)

⎤⎦ (2.111)

where εin, εout are the permittivities inside and outside the sphere. Next, we will eliminate allterms with summations in Equations 2.109 and 2.111 except one, by using the orthogonalityand normalisation properties of the Legendre polynomials:

∫ +1

–1Pl(cos θ)Pm(cos θ) d(cos θ) =

22m + 1

δ(ml) (2.112)

The result is

Amrm0

(2

2m + 1

)= –

23E0r0 + Bmr

–(m+1)0

(2

2m + 1

)(2.113)

and

Bmr–(m+1)0

(2

2m + 1

)[1 +

εout

εin

(m + 1m

)]=

23E0r0

[1 –

εout

εin· 1m

](2.114)

or,

Bmr–(m+1)0

(2

2m + 1

)=

23E0r0

[1 – εoutεin

· 1m]

[1 + εout

εin

(m+1m

)] (2.115)

Substituting Equation 2.115 into the right-hand side of Equation 2.113 results in the followingexpression involving Am and r0:

Amrm0

(2

2m + 1

)= –

23E0r0 +

23E0r0

[1 – εoutεin

· 1m]

[1 + εout

εin

(m+1m

)] (2.116)

The continuity condition should hold for any choice of r0, the radius of the sphere, and wesee that if we choose m = 1, then the expression for Am will be independent of r0. Then wefind


Example 3 continued

A1 = –E0

[3εoutεin+2εout

](2.117)

and from Equation 2.115

B1 = r30E0

[εin – εoutεin + 2εout

](2.118)

Now we can substitute A1 and B1 back into Equations 2.108 and 2.107 to obtain theexpressions for the scalar potential U(r, θ) outside and inside the dielectric sphere:

U(r, θ)out = –E0r cos θ + E0r30r2

cos θ[εin – εoutεin + 2εout

]outside potential (2.119)

U(r, θ)in = –E0r cos θ[

3εoutεin + 2εout

]inside potential (2.120)

From Equations 2.119 and 2.120 we can easily calculate the E-field outside and inside thesphere, which was the initial goal we set out to reach:

Ein(r, θ) =[

3εoutεin + 2εout

]E0ez (2.121)

Eout(r, θ) = E0z + E0


]r30r3(2 cos θ er + sin θ eθ

)(2.122)

2.6.1 Integral form of Gauss’s law

Maxwell’s equations are usually summarised in their differential form, but they canbe cast in an integral form, often more useful for practical calculation. The integralform of Equation 2.31 is found by applying Gauss’s theorem of vector calculus (seeAppendix C), ∫

V∇ ·K dτ =

∫SK · dσ (2.123)

This theorem states that the integral of the divergence of a vector field K over a volumeis equal to the integral of the field itself over a surface enclosing the volume. Applyingthis theorem to the differential form of Gauss’s law we have∫

V∇ ·D dτ = ε0

∫V

∇ · E dτ = q = ε0

∫SE · dσ (2.124)


q1

q2

qn

E1

En

E2

E3

dS

E

Figure 2.5 Charges q1, q2, q3, . . . are point sources on whichE-field lines originate. The integral form of Gauss’s law states thatthe surface integral of E over the volume enclosing these charges isequal to the total charge divided by the permittivity.

where q is the total charge enclosed by the volume, q =∫V ρ dτ . Note that, strictly

speaking, ρ in the integrand includes bound as well as free charge density. Figure 2.5illustrates the physical significance of the integral form of Gauss’s law.

2.7 Dynamic fields and potentials

2.7.1 Maxwell’s equations revisited

In Section 2.4 we wrote down Maxwell’s equations

∇ · E =ρfree

ε0

∇ ·B = 0

∇ × E = –∂B∂t

Faraday’s law

∇ ×H =∂D∂t

+ J Maxwell–Ampère law

In the case of static fields, the E-field is irrotational, ∇ × E = 0, and the B-fieldis ‘divergenceless’, ∇ · B = 0. Therefore, the E-field can be specified by the gradient ofa scalar function, the scalar potential V , and the B-field is determined by the curl of avector function, the vector potential A:

E = –∇V B = ∇ × A (2.125)

Dynamic fields and potentials 35

In the case of fields changing in time, the E-field is no longer irrotational because

∂B∂t

= ∇ × ∂A∂t

and substituting into Faraday’s law we have

∇ × E = –∇ × ∂A∂t

or

∇ ×(E +

∂A∂t

)= 0 (2.126)

The quantity E + ∂A/∂t is itself irrotational and can therefore be written as the gradientof a scalar function:

E +∂A∂t

= –∇V

or

E = –∇V –∂A∂t

(2.127)

Then, from Maxwell’s E-field divergence equation,

∇ · E = ∇ ·[–∇V –

∂A∂t

]=ρfree

ε0

or

∇2V +∂

∂t(∇ ·A) = –

ρfree

ε0(2.128)

Substituting B = ∇ × A into the left-hand side of the Maxwell–Ampère law and takingthe time derivative of Equation 2.127 we find

∇ × (∇ ×A) = μ0J – μ0ε0∇(∂V∂t

)– μ0ε0

∂2A∂t2

and using the vector field identity ∇ × (∇ ×A) = ∇(∇ ·A) – ∇2A, we can simplify to

(∇2A – μ0ε0

∂2A∂t2

)– ∇

(∇ ·A + μ0ε0

∂V∂t

)= –μ0J (2.129)


Equations 2.128 and 2.129 are just a reformulation of the Maxwell relations in terms ofthe time-dependent potentials rather than fields.

2.7.2 Lorentz gauge

The vector potential A can be adjusted to a certain extent since the B-field only dependson the curl of A, not A itself. In particular, since ∇× (∇f ) = 0, where f is some arbitraryscalar function, we can add the gradient of a scalar function to the vector potential with-out changing its curl. The choice of what scalar function to use is called the choice ofgauge, and one particularly useful choice of scalar function is called the Lorentz gauge.It is defined by

f = ∇ ·A = –μ0ε0∂V∂t

(2.130)

Substituting Equation 2.130 into Equation 2.129 we see that it simplifies to

∇2 ·A – μ0ε0∂2A∂t2

= –μ0J (2.131)

and Equation 2.128 becomes

∇2V – μ0ε0∂2V∂t2

= –ρfree

ε0(2.132)

The strategy then, given the source terms J and ρ, is to solve the two inhomogeneouswave equations, Equations 2.131 and 2.132, for A and V , and then use Equations 2.125and 2.127 to find the fields B and E.

2.7.3 Retarded potentials

The points in space where the potentials are evaluated may be very far from the loca-tion of the sources. Therefore, as the sources J(r′, t), ρ(r′, t) vary with time, the changesin the potentials at V (r, t), A(r, t) will only take effect at a later time t, governed bythe distance from the source position to the points in space where the potentials areevaluated, and the speed of light. The time that the changes actually take place at thesources is called the ‘retarded time’ tr , and the time at which these changes take effectat the potential points is just denoted by t. The relation between the two is tr = t – �/c,where � is the distance from the source point to the field point, � = |r – r′|. The spatialrelations between coordinate origin, source position, and field position are shown in Fig-ure 2.6. The formal solutions for distributed charge and current density sources, ρ(r′, tr),J(r′, tr) are

V (r, t) =1

4πε0

∫ρ(r′, tr)�

dτ ′ (2.133)

Dynamic fields and potentials 37

S

r

O

r

P

Source

Distribution

Figure 2.6 Relations between coordinate origin O,position r′ of differential volume dτ ′ in distributedsource, and distance � = |r – r′| between differen-tial source volume dτ ′, located at point S, and fieldpoint P.

and

A(r, t) =μ0

4π

∫J(r′, tr)�

dτ ′ (2.134)

Using these formal solutions for V and A, we can obtain formal solutions to the fieldsfrom Equations 2.127 and 2.125:

E(r, t) =1

4πε0

∫ [ρ(r′, tr)�2

� +ρ(r′, tr)c�

� –J(r′, tr)c2�

]dτ ′ (2.135)

and

B(r, t) =μ0

4π

∫ [J(r′, tr)�2

+J(r′, tr)c�

]× �dτ ′ (2.136)

In Equations 2.135 and 2.136, J denotes the time derivative of the source current density,and thus we write

∂A∂t

=μ0

4π

∫J�dτ ′ (2.137)

which is used in the third term on the right-hand side of Equation 2.135.


2.8 Dipole radiation

In this section, we use the results for the time-dependent retarded potentials developedin Section 2.7 to calculate the E-field and B-field solutions for a harmonically oscillatingdipole source. We take the dipole as two charges +q and –q placed at the ends of veryshort conducting wire of length a. The charges oscillate along the wire with a frequencyω such that the time dependence of the charge at any point along the wire is q(t) =q0 cos(ωt). The electric dipole, aligned along the z-axis, is then given by

p(t) = q(t)az = q0a cos(ωt)z = p0 cos(ωt)z (2.138)

The current i associated with this dipole is just

i(t) =dqdtz = –q0ω sinωtz (2.139)

Substituting the current into Equation 2.134, the vector potential for the harmonicoscillator is

A(r, t) =μ0

4π

∫ a/2

a/2

–q0ω sin[ω(t – r

c

)]z

rdz (2.140)

But since the length a of the oscillator is short compared to the wavelength, we cansimply replace the integral with the integrand multiplied by a:

A(r, t) � –μ0p0ω4πr

sin[ω(t –

rc

)]z (2.141)

We can write down the retarded scalar potential for some point r at time t, V (r, t) byusing Equation 2.133 and Figure 2.7:

V (r, t) =1

4πε0

[q0 cos [ω(t – r1/c)]

r1–q0 cos [ω(t – r2/c)]

r2

](2.142)

The denominators r1, r2 can be written in terms of r and the angle θ as

r1,2 =[r2 ∓ ra cos θ +

(a2

)2]1/2(2.143)

Now, if the point where the field is to be evaluated is sufficiently far from the dipole suchthat|r|� a, and the length of the dipole is sufficiently short such that the wavelength oflight λ = ω/c� a, then it can be easily shown, by expanding 1/r1,2 in a Taylor series, that

V (r, θ , t) � p0 cos θ4πε0

[–ω

rcsin[ω(t –

rc

)]+

1r2

cos[ω(t –

rc

)]](2.144)

Dipole radiation 39

r1

r2

P

r

a

−q

+q θ

ϕ Figure 2.7 Electric dipole oscillator alignedalong z-axis. The maximum length of the di-pole is a and the coordinates of field pointp, with respect to the dipole midpoint, areindicated in spherical coordinates r, θ ,φ.

In order to calculate the time-varying E-field, we need to know A(r, t) as well asV (r, θ , t) because

E = –∇V –∂A∂t

(2.145)

Using Equation 2.141 for the vector potential of a dipole aligned along the z-axis,

∂A∂t

= –μ0p0ω2

4πrcos

[ω(t –

rc

)]z (2.146)

expressed in spherical coordinates as

∂A∂t

= –μ0p0ω2

4πrcos

[ω(t –

rc

)](cos θ r – sin θ θ) (2.147)

Substituting Equations 2.144 and 2.147 into 2.145 and writing the gradient operator ∇in spherical coordinates (see Appendix D),

∇ =∂

∂rr +

1r∂V∂θ

θ

we find the radial component of the E-field

Er =2p0 cos θ4πε0

{1r3

cos[ω(t –

rc

)]–ω

r2csin[ω(t –

rc

)]}r (2.148)


and the angular component

Eθ =p0 sin θ4πε0

{1r3

cos[ω(t –

rc

)]–ω

r2csin

[ω(t –

rc

)]

–ω2

rc2cos

[ω(t –

rc

)]}θ (2.149)

Then, using B = ∇ × A we find, for the oscillator B-field,

Bϕ = –μ0p0 sin θ

4π

{ω

r2sin[ω(t –

rc

)]+ω2

rccos

[ω(t –

rc

)]}ϕ (2.150)

2.8.1 Far field

In the ‘far field’, where r� λ, the only surviving E-field term is

Eθ = –p0 sin θ4πε0

ω2

rc2cos

[ω(t –

rc

)]θ (2.151)

and the surviving B-field term is


4πω2

rccos

[ω(t –

rc

)]ϕ (2.152)

which can be rewritten, using μ0ε0 = 1/c2, as

Bϕ = –p0 sin θ4πε0

ω2

rc3cos

[ω(t –

rc

)]ϕ (2.153)

Comparing Equations 2.151 and 2.153, we see that the field amplitudes are relatedsimply by a factor of c,

Bϕ =1cEθ (2.154)

that the Eθ and Bϕ components are in phase, and that they are orthogonal to each otherand to the radial propagation direction r. The components Eθ and Bϕ in the far field asr→∞ become the plane waves discussed in Section 3.1.

The Poynting vector describes the energy flux emitted into the far-field region by theoscillating dipole:

S =1μ0

(E×B) =μ0p20ω

4

16π2c

(sin2 θ

r2

)cos2

[ω(t –

rc

)]r (2.155)

Light propagation in dielectric and conducting media 41

The optical-cycle-averaged energy flux 〈S〉 (energy per unit area per unit time) is

〈S〉 = μ0p20ω4

32π2c

(sin2 θr2

)(2.156)

and the total power W (energy per unit time) emitted by the oscillator is calculated byintegrating the energy flux over all space

W =∫〈S〉 · da =

μ0p20ω4

32π2c

∫sin2 θr2

r2 sin θdθdϕ =μ0p20ω

4

12πc(2.157)

2.8.2 Quasi-stationary regime

In the regime where r � λ, or equivalently where ωr/c � 2π , we can drop the retardationterm in the time arguments of Equations 2.148, 2.149, and 2.150, and rewrite them as

Er =2p0 cos θ4πε0r3

[cosωt –

(ωrc

)sinωt

]r (2.158)

Eθ =p0 sin θ4πε0r3

[cosωt –

(ωrc

)sinωt –

(ωrc

)2cosωt

]θ (2.159)

Bϕ = –p0 sin θ4πε0cr3

[(ωrc

)sinωt +

(ωrc

)2cosωt

]ϕ (2.160)

We see that, in the quasi-stationary regime, the surviving E-field and B-field terms froman oscillator source (the first terms on the right in Equations 2.158–2.160) are in quad-rature, not in phase as in far-field radiation. This regime corresponds to conventionalengineering circuit response where the product of frequencies ω (radio and microwave)and circuit size r are well within the ωr/c � 2π criterion.

Note that, when the quasi-stationary regime goes to the truly stationary, ω → 0,Equations 2.158–2.160 go to their electrostatic dipole limits,

Er =2p0 cos θ4πε0r3

Eθ =p0 sin θ4πε0r3

Bϕ = 0

2.9 Light propagation in dielectric and conducting media

So far we have assumed that light propagates either through a vacuum or through a gasso dilute that we need consider only the isolated field–particle interaction. Here we studythe propagation of light through a continuous dielectric (non-conducting) medium andnear the surface of a good conductor. Interaction of light with such media permits us tore-examine the important quantities of polarisation, magnetisation, susceptibility, indexof refraction, extinction coefficient, and absorption coefficient. We shall see later how the


polarisation field can be usefully regarded as a density of transition dipoles induced inthe dielectric by the oscillating light field. Formally, as the definitions Equations 2.17 and2.18 show, the magnetisation M of materials is on an equal footing with the polarisationP, but in practice, electric polarisation is encountered much more commonly. We willrestrict the present discussion, therefore, to the response of materials to the incidentelectric field component of a plane wave. We begin by recalling the definition of materialpolarisation P with respect to an applied electric field E as

P = ε0χeE (2.161)

where χe is the linear electric susceptibility: an intrinsic property of the medium respond-ing to the light field. Recall the relation between the electric field E, the polarisationP, and the displacement field D in a material medium. In the SI system of units therelation is

D = ε0E + P (2.162)

Furthermore, for isotropic materials, in all systems of units, the so-called ‘constitutiverelation’ between the displacement field D and the imposed electric field E, is written

D = εE (2.163)

with ε being referred to as the permittivity of the material. Therefore,

D = ε0(1 + χ)E (2.164)

and

ε = ε0(1 + χ) (2.165)

The susceptibility χ is often a strong function of frequency ω around resonances andcan be spatially anisotropic. It is a complex quantity, having a real dispersive componentχ ′ and an imaginary absorptive component χ ′′

χ = χ ′ + iχ ′′ (2.166)

Real and imaginary parts of the permittivity are related to the susceptibility by

ε = ε0(1 + χ ′) + iε0χ ′′ (2.167)

with

ε′ = ε0(1 + χ ′) (2.168)

ε′′ = ε0χ ′′ (2.169)

Light propagation in dielectric and conducting media 43

A number of familiar expressions in free space become modified in a dielectric medium.For example:

(kcω

)2

= 1 ; free space(kcω

)2

= 1 + χ ; dielectric

In a dielectric medium kc/ω becomes a complex quantity that is conventionally ex-pressed as

kcω

= η + iκ (2.170)

where η is the refractive index and κ is the extinction coefficient of the dielectric me-dium. The relations between the refractive index, the extinction coefficient, and the twocomponents of the susceptibility are

η2 – κ2 = 1 + χ ′ (2.171)

2ηκ = χ ′′ (2.172)

Note that, in a transparent lossless dielectric medium,

η2 – κ2 = 1 + χ ′ =ε′

ε0(2.173)

But if the medium has an absorptive component

2ηκ =ε′′

ε0(2.174)

The unitless terms ε′/ε0 and ε′′/ε0 are called the real and imaginary parts of the relativepermittivity or the dielectric constant. They are denoted as

ε′r =ε′

ε0(2.175)

ε′′r =ε′′

ε0(2.176)

The subscript r emphasizes relative permittivity. Unfortunately, in the scientific litera-ture the dielectric constant is often just written ε = ε′ + iε′′ with the same notationas the permittivity, and one must decipher from the context whether the permittivity


(units of C2/J · m) or relative permittivity–dielectric constant (unitless) is intended. Forexample, it is clear from expressions involving the refractive index, such as

n = η + iκ =√ε (2.177)

or a propagation parameter

k = k0n = k√ε (2.178)

that ε is the dielectric constant, whereas in an expression involving a constitutive relation,such as

D = εE (2.179)

the ε factor denotes a material permittivity. Note, however, that in the expression

D = εε0E (2.180)

The factor ε is the unitless dielectric constant (relative permittivity) and ε0 is the vacuumpermittivity.

In a dielectric medium the travelling wave solutions of Maxwell’s equation become

E = E0ei(kz–ωt) –→ E0e[iω(ηzc –t)–ω κc z] (2.181)

the relation between magnetic and electric field amplitudes is

B0 =√ε0μ0 E0 –→ B0 =

√ε0μ0 (η + iκ)E0

and the period-averaged field energy density is

ρω =12ε0 |E|2 –→ ρω =

12ε0η

2 |E|2 (2.182)

Now, the light-beam intensity in a dielectric medium is attenuated exponentially byabsorption:

Iω =12ε0c |E|2 –→ Iω =

12ε0η

2 |E|2(cη

)=

12ε0cηE2

0e–2 ωκc z = I0e–Kz (2.183)

where

I0 =12ε0cηE2

0 (2.184)

Exercises 45

is the intensity at the point where the light beam enters the medium, and

K = 2ωκ

c=ω

ηcχ ′′ (2.185)

is termed the absorption coefficient. Note that the energy flux Iω in the dielectric mediumis still the product of the energy density

ρω =12ε0η

2 |E|2 (2.186)

and the speed of propagation c/η. Note also that, although light propagating through adielectric maintains the same frequency as in vacuum, the wavelength contracts as

λ =c/ην

(2.187)

2.10 Summary

In this chapter we have passed in review many of the ideas, concepts, and quantities al-ready familiar to the reader as a warm-up exercise. We first discussed the basic forcefields E and B and the constitutive relations that join them to D and H. We thenintroduced Maxwell’s equations, the notion of bound charge, bound current, and thecharge-current continuity relation. We then introduced the expressions for field poten-tial, field energy, and energy transport by Poynting’s theorem.We then tried to concretisethese ideas with three illustrative examples of increasing complexity. The examplesare useful in themselves but they also showcase certain techniques and approaches tosolving problems commonly encountered in electrostatics. The focus then passed totime-dependent ‘dynamic’ solutions to Maxwell’s equations where the vector potentialand the Lorentz gauge were introduced. The reader then got a first taste of dipole radi-ation in the near field, the far field, and the ‘quasi-stationary regime’, before the chapterconcluded with some useful expressions for light propagation in dielectric media.

2.11 Exercises

1. A plane wave propagates in free space with an E-field and B-field given byEquations 3.14 and 3.16. Calculate the Poynting vector S associated with this wave.

2. As shown in Figure 2.8, a plane wave of wavelength 620 nm in air is incident fromthe left on a lossless, non-magnetic dielectric slab with dielectric constant ε′ = 2.25.Calculate the reflection and transmission coefficients R,T , the reflected power, andthe transmitted power assuming an E-field amplitude of 1V/m. Assume the dielectricslab to be infinitely thick.


incident wave

reflected wave

transmitted wave

z

x

ε0,μ0 ε ,μ0

Figure 2.8 Plane wave propagating from left to right, reflected atthe interface.

3. Now reconsider the dielectric slab in Figure 2.8 as a slightly lossy dielectric with anindex of refraction corresponding to SiO at λ0 = 620 nm. At this wavelength n =η + iκ = 1.969 + i0.01175. Calculate the reflection and transmission coefficients, thereflected power, and the transmitted power for this case.

4. Silicon nitride, Si3N4, is a very low loss dielectric material commonly used in micro-and nanofabrication. The index of refraction of silicon nitride in the visible re-gion of the spectrum is 2.05. Calculate the ratio of the impedance of a plane wavepropagating in Si3N4 to that of a plane wave propagating in free space.

5. The bulk plasma frequency of silver (Ag) metal is, ωp = 1.3× 1016 s–1. A plane wavewith an amplitude of 1 V/m, impinging on a silver surface, penetrates to a skin depthof about 5 nm. Calculate the wavelength of light and the current density Jc inducedin the Ag near the surface.

6. Consider a hydrogen atom subject to a constant, uniform electric field. Supposethe hydrogen atom consists of a positive point charge with a proton mass, sittingat the origin. A negatively charged uniform electron charge density cloud representsthe electron. The volume integral of the electron charge density is equal to the protoncharge. Because of the external E-field, the electron charge cloud is displaced fromthe proton. Use Coulomb’s law to calculate the restoring force between the chargecentres. Use this restoring force in a harmonic oscillator model to estimate a ‘natural’hydrogen atom frequency.

7. With the electron in the ground state of the hydrogen atom, the total energy of thesystem is known to be –13.6 eV.1 The total energy is the sum of the kinetic energy ofthe system in the ground state and the potential energy. If the kinetic energy of theelectron is 13.6 eV, what is the potential energy (due to the Coulomb attractive forcebetween the electron and proton)? Use your answer and the Planck relation E = hω

1 The unit of energy ‘eV’ is the electron volt and is the energy acquired by an electron after passing througha potential difference of 1 V. One eV is equivalent to 1.602× 10–19 J.

Further reading 47

to calculate the corresponding frequency ν. Compare this frequency to the harmonicoscillator frequency you calculated in Exercise 6.. Remember: the angular frequencyand cyclic frequency are related by ω = 2πν.

8. Redo Exercise 6 supposing that the applied E-field is harmonically oscillating withan angular frequency ω. (a) Estimate the average value of the oscillating dipole es-tablished between the positive and negative charge centres of the atom. (b) Calculatethe rate of energy (in Watts) emitted by the radiating dipole.

9. The ionosphere can be considered a charged-particle plasma gas. Calculate theplasma resonance frequency ωp. Assume that the density of free electrons is 105 cm–3.As we shall see in Chapter 7, according to the dispersion diagram for the charged-particle-plasma-gas model, waves can propagate into the plasma above ωp but arereflected below it. In radio transmission, above what frequency are we broadcastingto the solar system (and beyond)?

2.12 Further reading

1. D. J. Griffiths, Introduction to Electrodynamics, 3rd edition, Pearson Addison Wesley,Prentice Hall (1999).

2. J.-P. Pérez, R. Carles, and R. Fleckinger, Électromagnétism Fondements et applications,3ème édition, Masson (1997).

3. M. Born and E. Wolf, Principles of Optics, 6th edition, Pergamon Press (1980).

4. H. A. Haus,Waves and Fields in Optoelectronics, Prentice Hall (1984).

5. D. M. Pozar,Microwave Engineering, 3rd edition, John Wiley & Sons (2005).

6. M. Mansuripur, Field, Force, Energy and Momentum in Classical Electrodynamics,Bentham e-Books (2011).

7. F. Nunes, T. Vasconcelos, M. Bezerra, and J. Weiner, Electromagnetic energy densityin dispersive and dissipative media. Journal of the Optical Society of America B, vol 28,pp. 1544–52 (2011).

3

Physical Optics of Plane Waves

3.1 Plane electromagnetic waves

3.1.1 Phasor form of the vector fields

Up to now we have tacitly assumed that the vector force fields of electricity and mag-netism are real. However, for most of the following chapters we shall be studyingelectromagnetic waves of various sorts, propagating through dielectric and conductivemedia. The sources of these waves will be posited to be dipoles harmonically oscillatingat the angular frequency ω = 2πν and removed spatially to negative infinity. Therefore,the time dependence of the fields will always be set to a factor of e–iωt, and the spatialoscillation will always have an associated phase, eiϕ . The general form of the vector fieldscan then be expressed as

A(r, t) = A(r)eiϕe–iωt (3.1)

The use of A here should not be confused with the vector potential. Here, it is just ageneric vector field. If the spatial dependence of the wave is sinusoidal as well, then thespatial phase ϕ can be written as

ϕ = k · r + δ (3.2)

where k is called the propagation vector, the magnitude of which is related to thewavelength λ by

k =2πλ

(3.3)

and δ is just the reference phase at the wave origin, usually set to zero. The direction ofthe propagation vector is given by

k = kxx + kyy + kzz (3.4)


Plane electromagnetic waves 49

The form1, always expressed as that of the general vector field, Equation 3.1, is then

A(r, t) = A(r)ei(kxx+kyy+kzz)e–iωt (3.5)

The vector form of the amplitude A(r) reflects the fact that in most instances the amp-litudes components Ax,Ay,Az are not equal, and the amplitude is polarised linearly,circularly, or radially. In the subsequent discussion we will assume the simplest case oflinear polarisation with electromagnetic vector fields aligned along some cartesian axis.The direction of polarisation is described by vector e(r) of unit length so that we canwrite as

A(x, y, z, t) = e(r) ·A(r)ei(kxx+kyy+kzz)︸︷︷︸phasor

e–iωt (3.6)

and the complex spatial part of the field is called a phasor. Unless otherwise explicitlystated, we will assume from here forward that the usual fields of electromagnetism,E,B,D,H,P,M,J, and S are in the phasor form.

3.1.2 Decoupling Maxwell’s curl equations

We saw in Section 2.8.1 that the far-field spherical waves morphed asymptotically intolocal plane waves. We can also simply consider that the dipole source is removed to–∞ and posit the plane wave form as possible solutions to Maxwell’s equations. Thetwo curl equations, Equations 2.27 and 2.28, are at the heart of classical electrodynam-ics. They are two coupled first-order differential equations, but they can be uncoupledto form two second-order differential equations that admit propagating electromagneticwave solutions. The standard way to carry out the uncoupling is to apply the curl oper-ation to Equations 2.27 and 2.28, invoke a vector calculus identity, and posit source-freeconditions. The curl operations yield

∇ × (∇ × E) = ∇ (∇ · E) – ∇2E = ∇ ×(–∂B∂t

)(3.7)

= –∂

∂t(∇ ×B) = –μ0ε0

∂2E∂t2

(3.8)

1 The reader should be mindful that physics literature and engineering literature use different conventionsto describe time-harmonic phasor fields. The form A = A0ei(k·r–ωt) is the physics convention. The form A =A0e–j(β·r–ωt) is the engineering convention. An important consequence of this choice is that complex quantitiesa must be expressed as a = a + iα with the physics convention and a = a – jα with the engineering convention.Thus, a complex propagation parameter in the physics convention would be written k = k+ iκ, while engineerswould write β = α – jγ . Most of the circuit, transmission line, and waveguide literature is written by engineers.

50 Physical Optics of Plane Waves

In source-free regions ∇ · E = 0 so

∇2E = μ0ε0∂2E∂t2

=1c2∂2E∂t2

(3.9)

Similarly,

∇ × (∇ ×B) = ∇(∇ ·B) – ∇2B = ∇ ×(μ0ε0

∂E∂t

)(3.10)

= μ0ε0∂

∂t(∇ × E) = –μ0ε0

∂2B∂t2

(3.11)

According to Equation 2.26, the divergence of the magnetic induction field is alwaysnull, ∇ ·B = 0, so,

∇2B = μ0ε0∇2B∂t2

=1c2∇2B∂t2

(3.12)

Equations 3.9 and 3.12 clearly have solutions of the form of plane waves propagatingwith velocity c. We shall generally be concerned with light propagating in free space, orin dielectric and conductive materials. We start by describing the properties of the sim-plest form of light propagation: the plane electromagnetic wave composed of two fields,electric and magnetic, oscillating at a single frequency ω. Figure 3.1 shows the essential

Plane Phase

PropagationDirection

PropagationDirection

x

x

y

z

z

Magnetic Field

Electric Fieldy

k

k

λ

λ

Figure 3.1 Plane electromagnetic wave propagating in free space. The E-field is aligned alongx, the B-field along y, and the wave vector k is oriented along z.


features of the plane light wave. The electric field E constituent of the electromagneticfield, propagating in free space as a plane wave in direction k, is given by

E = eEE0ei(k·r–ωt) (3.13)

In particular, with propagation in the z direction, the wave has the form

E = eEE0ei(kzz–ωt) (3.14)

The complex field amplitude is E0, the polarisation direction eE, and the frequency ω.The frequency, wavelength λ, and velocity v of the constant phase front are related by

ω =2πλv = kv (3.15)

In free space v is the speed of light c, and when in other homogeneous materials, isgiven by

v =cn

where n is the index of refraction of the material. The index of refraction in free space isunity, and in matter, always greater. In materials with negligible absorption, n is a realnumber greater than unity, but in absorbing ‘lossy’ materials, n is complex with a positiveimaginary term. As light of a single frequency propagates through materials of differentindices of refraction, ω remains constant, but the phase velocity changes as c/n and thewavelength shortens by λ/n. It is important not to confuse the ‘angular’ frequency ω withthe frequency of optical cycles ν. The relation is always

ω = 2πν

The magnetic induction field B, the other constituent of the plane electromagnetic wavepropagating in free space in the z direction, is given by

B = eBB0ei(kzz–ωt) (3.16)

with complex amplitude B0 and polarisation direction eB. The reason for denoting thefield amplitudes complex is to express a possible phase relation between them. In fact,the divergence and curl relations in Maxwell’s equations ensure that the E- and B-fieldsare orthogonal to the direction of propagation and that the amplitudes are in phase. TheE- and B-fields combine to form the electromagnetic (E-M) field. They are mutuallyorthogonal and orthogonal to the direction of propagation:

eE · eB = 0 and eE × eB = ek


where ek is the unit vector of the propagation parameter k. The relative amplitudebetween magnetic induction and electric fields in vacuum is given by,

B0 =1cE0 =

√ε0μ0E0 (3.17)

and for a travelling wave through free space and homogeneous matter, the timeoscillation of the E- and B-fields is always in phase.

3.1.3 Plane wave field relations from Maxwell’s equations

For plane waves with E-field and H-field in the x – y plane, propagating along z in a di-electric medium with permittivity ε and permeability μ, and oscillating at frequencyω, we have, from the curl relations of Faraday’s law and the Maxwell–Ampère law,Equations 2.27 and 2.28;

∂Ex∂z

= –μ∂Hy

∂t= iωμHy (3.18)

∂Ey∂z

= μ∂Hx

∂t= –iωμHx (3.19)

and

∂Hy

∂z= –ε

∂Ex∂t

= iωεEx (3.20)

∂Hx

∂z= ε

∂Ey∂t

= –iωεEy (3.21)

Equations 3.18 and 3.20, relating Ex andHy, are coupled, as are Equations 3.19 and 3.21,relating Ey and Hx. The first of these pairs can be decoupled by differentiating Equa-tion 3.18 by z and Equation 3.20 by t. The order of differentiation is interchangeable,which results in

∂2Ex∂z2

– με∂2Ex∂t2

= 0 (3.22)

Taking into account the harmonic time variation at frequency ω, we have

∂2Ex∂z2

+ μεω2E2x = 0 (3.23)

The product of the material parameters is με = 1/v2, where v is the phase velocity ofthe plane wave in the medium. The plane wave propagation parameter is k = ω/v so


Equation 3.23 can be written finally as the one-dimensional wave equation for the Excomponent of the plane wave:

∂2Ex∂z2

+ k2E2x = 0 (3.24)

Clearly, analogous equations can be written for all the field components in the rela-tions Equations 3.18–3.21. Furthermore, substitution of the plane wave E-field solution,Equation 3.14, into the second-order wave equation, Equation 3.9, results in

∇2E + k2E = 0 (3.25)

with k = ω/c = ω√μ0ε0 when the waves are propagating in free space. Equation 3.25 is

clearly valid for each of the Cartesian components and can be written simply in terms ofthe wave amplitude,

∇2E + k2E = 0 (3.26)

and as such it is called the Helmholtz scalar wave equation. The analogous equationis obviously valid for the plane-wave H-field, and the relation between the E-field andH-field plane-wave amplitudes is specified by Equation 3.19:

B =Ec

and therefore E =√μ0

ε0H (3.27)

The proportionality factor between the amplitudes,√μ0/ε0, has units of impedance and

is variously called the impedance of free space, or intrinsic impedance, or wave impedanceand is denoted by Z0:

Z0 =√μ0

ε0= 376.73 Ohms (3.28)

In lossless media μ and ε are real numbers, but in lossy dielectrics or conductive mater-ials such as metals or semiconductors, the permittivity becomes complex. We examinethe interaction of plane waves and lossy or conductive media in the next section.

3.1.4 Plane waves in a lossy, conductive medium (lowfrequency limit)

In this section we examine the propagation of plane waves in a lossy conductor. Theconstitutive relations for permeability, permittivity, and conductivity, μ, ε, σ , respect-ively, characterise the medium. As usual, we do not consider ‘magnetic’ materials so thepermeability remains μ0 even inside the conductor, but the permittivity becomes com-plex, ε = ε0(ε′ + iε′′). Here we assume that σ is real, which is perfectly adequate for RF


and microwave frequencies. Later in Section 3.2.4 we shall see that at optical frequenciesthe conductivity must also be generalised to a complex quantity.

We start again with Faraday’s law and the Maxwell–Ampère law including both dis-placement and charge current in a medium characterised by μ, ε, σ . Assume further thatOhm’s law obtains so that J = σE:

∇ × E = –∂B∂t

= –μ∂H∂t

∇ ×H =∂D∂t

+ J = ε∂E∂t

+ σE

Let us posit plane solutions with the E-field aligned along x, B-field aligned along y, andpropagating along z:

Ex = E0ei(kzz–ωt)

By = B0ei(kzz–ωt)(3.29)

The single frequency, harmonic time dependence results in

∇ × E = iμωH (3.30)

∇ ×H = (–iεω + σ )E (3.31)

Take the curl of both sides of Equation 3.30 and apply the standard ‘curl-curl’ identity

∇ ×∇ × E = iμω∇ ×H (3.32)

∇(∇ · E) – ∇2E = iμω∇ ×H (3.33)

–∇2E = iμω∇ ×H (3.34)

Because, as usual, we assume a source-free environment where the plane wave is propa-gating, ∇ ·E evaluates to zero in Equation 3.33. Now substituting Equation 3.31 into theright-hand side of Equation 3.34 results in

∇2E + μεω2(1 + i

σ

ωε

)E = 0 (3.35)

This is clearly a ‘Helmholtz-like’ wave equation, and since we have posited the E-fieldalong x,

∂2Ex∂z2

+ μεω2(1 + i

σ

ωε

)Ex = 0 (3.36)

and the H-field along y

∂2Hy

∂z2+ μεω2

(1 + i

σ

ωε

)Hy = 0 (3.37)


The k of Equation 3.26, assumed real, generalises to a complex expression

k =√μεω

(1 + i

σ

ωε

)1/2(3.38)

For non-magnetic lossy materials we can write μr = 1 and ε = ε0εr = ε0(ε′ + ε′′). Then:

k =2πλ0

[ε′ + i

(ε′′ +

σ

ωε0

)]1/2(3.39)

3.1.4.1 Plane waves in a lossy dielectric (but poor conductor)

If σ /ωε0 ε′′ then the material is considered a poor conductor or good insulator and thepropagation parameter becomes

k→ 2πλ0

(ε′ + iε′′)1/2 (3.40)

A plane wave propagating through this medium reflects dissipative loss due to ε′′:

Ex = E0ei[2π /λ0(ε′+iε′′)1/2z–ωt

](3.41)

In many cases∣∣ε′′/ε′∣∣ 1 in which case we can write

k � 2πλ0

√ε′(1 + i

ε′′

2ε′

)(3.42)

and Equation 3.41 can be written as

Ex = E0e–(2π /λ0)

ε′′2√ε′ zei[(2π /λ0)

√ε′z–ωt] (3.43)

The first exponential factor with the real argument damps the amplitude E0 as the wavepropagates along z. The second exponential factor represents the usual harmonic waveoscillation through a medium characterised as

√ε′. The two factors evidently represent

a damped travelling plane wave. As ε′′ → 0, the damping vanishes and the oscillatoryterm represents a plane wave propagating through a lossless dielectric with real refract-ive index η =

√ε′. The wave impedance in a lossy medium is again determined from

Faraday’s law. We find in this case:

Ex =√

μ0

ε0(ε′ + iε′′)Hy (3.44)


and

Z =√

μ0

ε0(ε′ + iε′′)(3.45)

In the limit of a lossless, non-magnetic dielectric, ε′′ → 0,

Z =

√μ0

ε= Z0

1√ε′

(3.46)

3.1.4.2 Plane waves in a good conductor (low frequency limit)

As in Section 3.1.4, here we continue to treat σ as a real quantity. We shall see in Section3.2.4 that although this assumption is adequate for radio and microwave frequencies, itbreaks down at optical frequencies.

When σ /ωε0� ε′′ we have a good conductor. The generalised propagation parameter,Equation 3.39 becomes, as ε′′ωε0/σ → 0,

k→ 2πλ0

(ε′ + i

σ

ωε0

)1/2

(3.47)

The good conductor (often a metal such as silver or gold) is then characterised by the realpart of the permittivity and the conductivity of the metal at frequency ω. If the dissipativepart ε′′ of the permittivity is retained, then the imaginary terms in Equation 3.39 can begrouped together and labelled as a ‘total’ ε′′tot:

ε′′tot = ε′′ +

σ

ωε0(3.48)

and an ‘equivalent conductivity’ including both terms of the imaginary part of thepermittivity can be defined as

σequiv = ωε0ε′′tot (3.49)

Returning to Equation 3.172, we see that if σ /ωε0� ε′ then a plane wave entering theconductor will by damped by the conductivity term:

k � 2πλ0

√i(σ

ωε0

)1/2

(3.50)

Using the identity√i = (1 + i)/

√2,

k � 2πλ0

(1 + i)(

σ

2ωε0

)1/2

(3.51)


The imaginary term damps the plane wave propagating into the conductor. The depthof penetration, the skin depth δ, is defined as inverse of this imaginary term

δ =λ0

2π

(2ωε0σ

)1/2

(3.52)

As a specific example, take silver metal as a good conductor and λ0 = 514.5 nm, the‘green line’ of an Argon-ion laser. At this wavelength, ε′ � –15 and ε′′ � 0.3. Theconductivity of silver is σ � 6.3× 107 S/m. The ratio σ /ωε0 � 2× 103 is much greaterthan the real or imaginary parts of the dielectric constant. The expression for the skindepth is therefore valid, and for this example, δ � 2.6 × 10–9 m. At optical frequenciesan incident plane only penetrates a few nanometres below the surface of a good metal.

The impedance is again determined from Faraday’s law, specifying the relationbetween the E- and H-fields in the conductor:

Ex =√μ0

ε0

[(1 + i)√

2

]–1√ωε0

σHy (3.53)

and

Z = Z0e–i(π /2)√ωε0

σ(3.54)

Note that the E-field lags the H-field in the conductor by a phase angle of 45 degrees.

3.1.5 Energy density and flux in harmonic phasor fields

The discussion of energy density and power in Section 2.5 must now be extended toaccount for the complex form of harmonic phasor fields. The expressions for the E-fieldenergy and B-field energy (Equations 2.43 and 2.49) in the static case and in vacuum are

Eelec =12

∫E ·D dτ

Emag =12

∫H ·B dτ

If the fields are time-harmonic phasors then the optical-cycle-averaged expressions forthe electric and magnetic field energies in vacuum are

Eelec =ε0

4

∫E · E∗ dτ (3.55)

Emag =μ0

4

∫H ·H∗ dτ (3.56)


From the relation between the amplitudes of the electric and magnetic fields

|H| = 1μ0c|E| (3.57)

It is clear that in vacuum

Emag = Eelec (3.58)

the magnetic field energy of the wave is equal to the electric field energy. Starting with anexpression analogous to Equation 2.53 we can develop a power balance relation betweenwork done on moving charges and the energy flow from plane-wave phasor fields. In thiscase we will include charges flowing in conductors Jc as well as freely moving chargedparticles Jfree and write the total current as

Jtot = Jfree + Jc (3.59)

and the time rate of doing work on, or by, these charges by the electromagnetic field is

dEdt

= E · Jtot (3.60)

Now we use the Maxwell–Ampère law and Faraday’s law in their forms appropriate fortime-harmonic phasors. We generalise the permittivity to include the possibility of lossydielectrics and conductors and write ε = ε0(ε′ + iε′′), but do not consider the muchless frequent case of magnetic materials. Therefore, we leave the permeability as μ0 andwrite

∇ ×H∗ = ε∗∂E∗

∂t+ J∗tot = iωε∗E∗ + J∗tot (3.61)

∇ × E = –μ0∂H∂t

= iωμ0H (3.62)

Now we ‘dot-multiply’ the conjugate Faraday’s law by E and the Maxwell–Ampère lawby H∗ and invoke the vector field identity

∇ · (E×H∗) = (∇ × E) ·H∗ – (∇ ×H∗) · E (3.63)

Substituting the result of the dot-multiplications into Equation 3.63 we find

∇ · (E×H∗) = iω(μ0|H |2 – ε0ε′|E|2

)– ωε0ε′′|E|2 – J∗free · E – σ |E|2 (3.64)

Plane wave reflection and refraction 59

Then applying Stokes’ theorem to the term on the left, rearranging the terms on theright, and dividing all terms by 2,

–12

∫J∗free · E dV =

12

∮S

(E×H∗

) · da +12

∫ωε0ε

′′|E|2 dV +12

∫σ |E|2 dV

+iω2

[∫ (ε0ε′|E|2 – μ0|H |2

)dV]

(3.65)

This power balance equation requires careful interpretation. The term on the left can beconsidered a source term from which energy is emitted. The negative sign denotes powerflowing away from the emitter through the surface S of the enclosing volume V . The firstterm on the right is immediately recognisable as the cycle-averaged power passing outof the volume enclosed by surface S. This term is the phasor version of Equations 2.58and 2.60, and we can identify the integrand with the Poynting vector

S = E×H∗ (3.66)

The next two terms indicate dissipative loss: absorptive loss from ε′′ and conductiveloss from σ due to material present within the enclosed volume. The last term on theright describes energy ‘stored’ in the material. Taking into account Equation 3.57, wecan write this last term as

dEstored

dt=iω2

[∫χ ′ε0|E|2

]dV = –

12

[∫E · dP

dt

]dV (3.67)

and we see that this term represents power flow into the polarisation of the material.Since polarisation is a density of dipoles, we can think of this term as that part of thetotal emitted energy that is stored in the dipole oscillators of the material rather than thedipole oscillators of the field. The negative sign indicates that the stored energy and fieldenergy oscillate out of phase.

3.2 Plane wave reflection and refraction

In this section we follow, except for minor variations, the development of Born andWolf, Principles of Optics (6th edition) because the present author could not see any wayto improve on it.

In general, a plane wave incident at an interface between two media with differentindices of refraction produces a reflected wave and a transmitted wave. These two newwaves will propagate away from the interface at angles and with amplitudes to be de-termined. The three waves propagate in space and in time as ei(k·r–ωt). At their commonpoint on the z = 0 interface (Figure 3.2), the time phase factor for all three is the sameand for all subsequent times this phase factor will evolve as t – (r · kqn/vn), where n = 1, 2,


the index of the medium, v1,2 the phase velocity in the medium, and q = i, r, t, theincident, reflected, or transmitted wave. Therefore, we can write

t –r · ki1v1

= t –r · kr1v1

= t –r · kt2v2

(3.68)

From Figure 3.2 we write any point along the interface r = xx+ zz and write out the dotproducts explicitly

xkix + zkiz

v1=xkrx + zk

rz

v1=xktx + zk

tz

v2(3.69)

Since Equation 3.69 must hold for any coordinates x, z along the interface, we can write

kixv1

=krxv1

=ktxv2

andkizv1

=krzv1

=ktzv2

(3.70)

The incident wave unit vector kiand a line normal to the interface (any line parallel to

the z-axis) form the plane of incidence. Equation 3.70 shows that the incident, reflected,and transmitted waves lie in this plane. We take this plane to be the X – Z plane as inFigure 3.2. Then all y components are null and the unit wave vector components can bewritten in terms of the angles θi , θr , θt,

kix = sin θi kiz = cos θi

krx = sin θr krz = cos θr (3.71)

ktx = sin θt krz = cos θt

We choose the incident wave to be travelling towards z = 0 from the negative z half-space. Since the reflected wave is travelling away from z = 0 in the same half-space, thez component of the reflected wave must be negative. From the foregoing (Equations 3.70and 3.71) we have,

sin θr = sin θi but cos θr = – cos θi (3.72)

So θr = π – θi as indicated in Figure 3.2. Equations 3.70 and 3.72 constitute the Law ofReflection: the sin of the angle of incidence and the sin of the angle of reflection are equaland in the same plane. We can also see from Equations 3.70 and 3.72, and rememberingthat the phase velocity v = c/n,

sin θisin θt

=v1v2

=n2n1

(3.73)


Equation 3.73, together with the statement that incident and transmitted propagationangles are in a plane, constitute the Law of Refraction or Snell’s law.

Fresnel theory

The laws of reflection and refraction determine propagation angles, but the amplitude,polarisation, and phase of these fields are also of interest. Expressions first derived byAugustin-Jean Fresnel (1788–1827) characterise these properties. We first focus on theamplitude of the incident plane-wave force field, which we write in the general phasorform of Equation 3.1:

A(r, t) = A(r)eiϕe–iωt (3.74)

where A(r, t) is either the E-field or H-field of the three plane waves, incident, re-flected, or transmitted. Next we group the amplitude and its spatially varying phaseinto a complex amplitude, A = A(r)eiϕ and write the time-varying phase iωt as inEquation 3.68.

τi = t –r · kiv1

= t –x sin θi + z cos θi

v1(3.75)

The complex amplitudes of all three waves are resolved into components parallel andperpendicular to the plane of incidence as shown in Figure 3.2. We then write thecartesian components of the incident E-field as

E(i)x = –E‖ cos θie–iωτi , E(i)

y = E⊥e–iωτi , E(i)z = E‖ sin θie–iωτi (3.76)

n2

n1

Z

X

T‖

E‖ R‖

T┴

R┴E┴

θt

θi

θr

Figure 3.2 Plane wave amplitudes, components par-allel and perpendicular to the plane of incidence, forthe incident E-field (E‖,E⊥), reflected E-field (R‖,R⊥),and transmitted E-field (T‖,T⊥) and angles θi,r,t of in-cidence, reflection, and transmission. The three wavesscatter in the X – Z plane at the interface between theindices of refraction n1, n2. The materials are supposedlossless, non-dispersive, with n2 > n1.


We find the cartesian components of the H-field by using the right-hand orthogonalityrelations between E and H:

H =√ε

μk × E (3.77)

We are usually not concerned with magnetic materials and set μ = 1. Then the incidentH-field cartesian components are

H (i)x = –

√ε1E⊥ cos θie–iωτi , H (i)

y = –√ε1E‖e–iωτi , H (i)

z =√ε1E⊥ sin θie–iωτi

(3.78)

Now we write the complex amplitudes of the reflected and transmitted fields as R and Tand their cartesian components as

E(r)x = –R‖ cos θre–iωτr , E(r)

y = R⊥e–iωτr , E(r)z = R‖ sin θre–iωτr (3.79)

and

H (r)x = –

√ε1R⊥ cos θre–iωτr , H (r)

y = –√ε1R‖e–iωτr , H (r)

z =√ε1R⊥ sin θre–iωτr

(3.80)where

τr = t –r · krv1

= t –x sin θr + z cos θr

v1(3.81)

For the transmitted E-field

E(t)x = –T‖ cos θte–iωτt , E(t)

y = T⊥e–iωτt , E(t)z = T‖ sin θte–iωτt (3.82)

and the transmitted H-field

H (t)x = –

√ε2T⊥ cos θte–iωτt , H (t)

y = –√ε2T‖e–iωτt , H (t)

z =√ε2T⊥ sin θte–iωτt

(3.83)with

τt = t –r · ktv2

= t –x sin θt + z cos θt

v2(3.84)

Field continuity conditions at the interface dictate that the tangential (x and y)components of the three fields must be continuous,

E(i)x + E(r)

x = E(t)x E(i)

y + E(r)y = E(t)

y (3.85)

H (i)x +H (r)

x = H (t)x H (i)

y +H (r)y = E(t)

y (3.86)


and substituting Equations 3.76, 3.78, 3.79, 3.80, 3.82 and 3.83 into 3.85 and 3.86, whileremembering that cos θr = cos(π – θi) = – cos θi , we have

cos θi(E‖ – R‖) = cos θtT‖ (3.87)

E⊥ + R⊥ = T⊥ (3.88)√ε1 cos θi(E⊥ – R⊥) =

√ε2 cos θtT⊥ (3.89)

√ε1(E‖ + R‖) =

√ε2T‖ (3.90)

We note that Equations 3.87 and 3.90 relate only components parallel to, and Equa-tions 3.88 and 3.89 only components perpendicular to, the plane of incidence. The twoequations involving parallel components are coupled, and the two equations involvingperpendicular components are coupled, but the two sets are independent. Componentsparallel and perpendicular to the plane of incidence are said to have ‘p-polarisation’and ‘s-polarisation’, respectively. The use of the terms p- and s-polarisation come fromthe German parallel and senkrecht. The two sets of relations are also referred to as TM(transverse magnetic) and TE (transverse electric) polarisations. We can take linear com-binations of Equations 3.88 and 3.89 to obtain expressions for R⊥ and T⊥ in terms ofthe incident E-field:

R⊥ =n1 cos θi – n2 cos θin1 cos θt + n2 cos θt

E⊥ (3.91)

T⊥ =2n1 cos θi

n1 cos θi + n2 cos θtE⊥ (3.92)

Similarly, linear combinations of Equations 3.87 and 3.90 result in

R‖ =n2 cos θi – n1 cos θtn2 cos θi + n1 cos θt

E‖ (3.93)

T‖ =2n1 cos θi

n2 cos θi + n1 cos θtE‖ (3.94)

These expressions for the reflected and transmitted amplitudes in terms of the incidentE-field amplitude are called the Fresnel relations. They can be recast2 , using the law of

2 The unitless expressions in brackets in Equations 3.95–3.98 and in parentheses in Equations 3.99–3.102are often referred to as the Fresnel coefficients, r, t. We will make extensive use of them in the subsequentdiscussion of reflection and refraction at a material interface. They are not to be confused with R,T whichhave field amplitude units.


refraction, Equation 3.73, and standard trigonometry identities, to eliminate n1, n2,

R⊥ = –{sin(θi – θt)sin(θi + θt)

}E⊥ (3.95)

T⊥ ={2 sin θt cos θisin(θi + θt)

}E⊥ (3.96)

R‖ ={tan(θi – θt)tan(θi + θt)

}E‖ (3.97)

T‖ ={

2 sin θt cos θisin(θi + θt) cos(θi – θt)

}E‖ (3.98)

For the common case of normal incidence the expressions Equations 3.91–3.94simplify to,

R⊥ = –(n – 1n + 1

)E⊥ (3.99)

T⊥ =(

2n + 1

)E⊥ (3.100)

R‖ =(n – 1n + 1

)E‖ (3.101)

T‖ =(

2n + 1

)E‖ (3.102)

where n = n2/n1. Note that for s-polarisation the Fresnel coefficients have the propertythat 1 = t⊥ – r⊥. This relation does not hold for p-polarisation, but 1 = r‖ + t‖ does holdat normal incidence.

3.2.1 Plane wave power reflection and refraction

In Section 3.2 we were concerned with plane wave amplitude reflection and transmissionat a boundary between two materials with different indices of refraction. Continuity ofthe tangential field components at the boundary lead to the laws of reflection and refrac-tion, and the Fresnel relations. In this section we consider reflection and transmission ofpower at the boundary. In addition to tangential field continuity we make use of energyconservation between incident flux normal to the boundary and the sum of reflected andtransmitted fluxes. From Equations 3.27, 3.46, and 3.66 we write the incident energy fluxor power density (J/m2s) as

S =√ε1ε0c |E|2 (3.103)

and the incident flux normal to the boundary is

F (i) = S cos θi =√ε1ε0c |E|2 cos θi (3.104)


The energy fluxes reflected and transmitted normal to the boundary are given by

F (r) =√ε1ε0c |R|2 cos θi = n1ε0c |R|2 cos θi (3.105)

F (t) =√ε2ε0c |T |2 cos θt = n2ε0c |T |2 cos θ2 (3.106)

The reflectivity and transmissivity are defined3 as the fractional reflected and transmittedfluxes,

R =F (r)

F (i)=|R|2|E|2 T =

F (t)

F (i)=n2n1

cos θtcos θi

|T |2|E|2 (3.107)

Invoking energy conservation across the boundary,

1 = R + T (3.108)

Now we take a closer look at the reflectivity and transmissivity as a function of incidentpolarisation. Let ϕ be the angle of the incident plane wave E-field with respect to theplane of incidence as shown in Figure 3.3. Then we have F‖ and F⊥, the energy fluxesparallel and perpendicular to the plane of incidence, expressed as

F (i)‖ = F (i) cos2 ϕi (3.109)

F (i)⊥ = F (i) sin2 ϕi (3.110)

and similarly for the reflected and transmitted fluxes. The reflectivity and transmissiv-ity can then be resolved into components parallel and perpendicular to the plane ofincidence,

R = R‖ cos2 ϕi + R⊥ sin2 ϕi (3.111)

n1

Y

E Z

X

n2

kiθi

φiFigure 3.3 Boundary in the X –Y plane betweentwo regions of indices of refraction n1 and n2. Theplane of incidence is formed by the incoming wavevector ki and the normal to the boundary along thez-axis.The angle θi is the angle of incidence betweenthe incoming wave and the normal to the bound-ary. The angle ϕi lies between the incident linearlypolarised E-field and the plane of incidence.

3 These quantities are also known as the reflectance and the transmittance.


where

R‖ =F (r)‖

F (i)‖

=|R‖|2|E‖|2 (3.112)

R⊥ =F (r)⊥

F (i)⊥

=|R⊥|2|E⊥|2 (3.113)

and

T‖ =F (t)‖

F (i)‖

=n2n1

cos θtcos θi

|T‖|2|E‖|2 (3.114)

T⊥ =F (t)⊥

F (i)⊥

=n2n1

cos θtcos θi

|T⊥|2|E⊥|2 (3.115)

From the expressions for the Fresnel amplitude relations, Equations 3.95–3.98 andEquations 3.114 and 3.115 the components of R and T can be written as

R‖ ={tan(θi – θt)tan(θi + θt)

}2

(3.116)

R⊥ ={

sin(θi – θt)sin θi + sin θt

}2

(3.117)

and

T‖ =sin 2θi sin 2θt

sin2(θi + θt) cos2(θi – θt)(3.118)

T⊥ =sin 2θi sin 2θtsin2(θi + θt)

(3.119)

Finally at normal incidence:

R ={n – 1n + 1

}2

(3.120)

T =4n

(n + 1)2(3.121)

It can be easily verified that R⊥+T⊥ = 1 and that R‖+T‖ = 1. In the special case whereθi + θt = π /2, Equation 3.116 shows that the parallel component of the reflected powergoes to zero. Furthermore, it can be seen from Figure 3.2, 3.3, and 3.4 that if the sum


Z

X

θi θic

θtc

n1

n2Figure 3.4 Total internal reflection at theinterface between two media with refractive indi-ces n1 > n2. At the critical angle of incidence θ cithe transmitted wave travels as a surface wavealong the interface boundary. The transmis-sion angle θ ct is π /2. At incident angles greaterthan θ ci total reflection occurs and no power istransmitted to the n2 half-space.

of the angles of incidence and transmission angles is 90 degrees, then the angle betweenthem must also be 90 degrees. Then the law of refraction reads

n2n1

=sin θisin θt

=sin θi

sin(π /2 – θi)=

sin θicos θi

= tan θi (3.122)

This special angle of incidence is called the Brewster angle. For an air/glass interface withthe refractive index of glass equal to 1.5, the Brewster angle is θi = tan–1(1.5) � 56degrees.

3.2.2 Total internal reflection

Consider again the law of refraction (Snell’s law).

n2n1

=sin θisin θt

or sin θt =n1n2

sin θi (3.123)

So far we assumed n1 < n2, the usual case when light in vacuum or in air impinges ona common isotropic dielectric material such as glass or fused quartz. Suppose, however,that the light is incident on the boundary from the high-index side so that n1 > n2. It isthen possible that above some incident angle θi Snell’s law yields a value for sin θt greaterthan unity. The value of θi such that sin θt is just equal to unity is called the critical angleθ ci and θt = π /2. As we shall show below, at angles θi > θ ci , there is no power transmissionand all the propagating flux is reflected at the boundary. When θi > θ ci , cos θt becomescomplex as can be seen from

cos θt =√1 – sin2 θt =

√1 –

(n1n2

)2

sin2 θi = ±i√(

n1n2

)2

sin2 θi – 1 (3.124)

Although no power is transmitted to the low-index side of the interface, Equa-tions 3.82, 3.83 and 3.84 show that evanescent E- and H-fields do exist there. For


convenience we rewrite these equations here for the transmitted E-field componentsand substitute Equations 3.123 and 3.124 for sin θt and cos θt:

E(t)x = –T‖ cos θte–iωτt , E(t)

y = T⊥e–iωτt , E(t)z = T‖ sin θte–iωτt (3.125)

where τt is given by

τt = t –r · ktv2

= t –x sin θt + z cos θt

v2(3.126)

τt = t –

(n1n2

sin θi)x +∓i

(√(n1n2

)2sin2 θi – 1

)z

v2(3.127)

Clearly the argument of the phase of the transmitted E-field components in the +z dir-ection (above the interface in Figure 3.4) becomes a real negative number if the negativeroot of the imaginary term in Equation 3.127 is chosen; the field amplitudes fall off ex-ponentially as +z increases. Choosing the positive root results in exponential fall-off inthe –z direction. At the critical angle, the wave does propagate along x at the boundarybecause the x-component of the phase term in Equation 3.127 is unity.

We can show that beyond the critical angle, all the light energy is reflected by using theFresnel relations, Equations 3.93 and 3.94, and substituting Equation 3.124 for cos θt.Since cos θt is purely imaginary, taking the absolute value of R‖ and R⊥ results in

|R|2‖ = |E|2‖ and |R|2⊥ = |E|2⊥ (3.128)

So the sum of the incident energy parallel and perpendicular to the plane of incident istotally reflected at the boundary.

We can also calculate the time-averaged energy flux across the boundary in the zdirection by evaluating the z-component of the Poynting vector. From Equations 3.82and 3.83 we form Sz from the relevant components of the E- and H-fields:

|S| = E×H∗ (3.129)

S(t)z = E(t)

x H(t)y – E(t)

y H(t)x (3.130)

We want to evaluate the time-average of the real part of Sz at the boundary z = 0:

Re[Sz(z = 0)] =12[Sz + S∗z] (3.131)


Substituting the relevant terms from Equations 3.82 and 3.83 into Equations 3.130 and3.131 we find

Re[Sz] =12

[(T2‖√ε2 cos θte–2iωτ + T2

⊥√ε2 cos θte–2iωτ

)+(

T∗2‖√ε2 cos∗ θte2iωτ + T∗2⊥

√ε2 cos∗ θte2iωτ

)](3.132)

and with cos θt pure imaginary when θi > θ ci ,

Re[Sz] =12

{√ε2 cos θt

[(T2‖ + T

2⊥)e–2iωτ –

(T∗2‖ + T∗2⊥

)e2iωτ

]}(3.133)

where, at z = 0,

τ = t –x sin θtv2

= t –n1n2

x sin θiv2

(3.134)

The four terms in Equation 3.133 simply oscillate harmonically in t with period T =1/2ω. We take the time average over a time t′ long compared to T :

〈Re[Sz]〉 = 12t′

∫ t′

–t′Re[Sz(t)]dt′ (3.135)

and find that

limt′→∞〈Re[Sz]〉 → 0 (3.136)

Therefore at times long compared to an optical period, no energy flux crosses theboundary when θi ≥ θ ci .

3.2.3 Reflection and transmission at a material interface

Reflection and transmission of light at surfaces is encountered frequently in the studyof light–matter interaction. We discuss here a few simple cases in which a plane wave,incident from a half-space of vacuum (ε0,μ0, σ = 0), impinges at the surface of a ma-terial characterised by ε,μ, σ . The cases are organised according to the nature (real orcomplex) and magnitude of the permittivity and conductivity. As before, we considerthe material non-magnetic and keep the permeability as μ0.

3.2.3.1 Normal incidence on a general material

We posit a plane wave propagating from left to right along z, with E-field linearly polar-ised along x (s-polarisation) and H-field along y as shown in Figure 3.1. The interfaceis in the x – y plane and situated at z = 0. The half-space to the left (z < 0) is vacuum,


and the half-space to the right (z > 0) is characterised by ε,μ0, σ . The incident E- andH-phasor fields, parallel to the x – y plane, are given by

Ei = E0eik0z x (3.137)

Hi = H0eik0z y =E0

Z0eik0z y (3.138)

where Z0 is the impedance of free space, Z0 =√μ0/ε0. At the interface, a fraction of

the wave is reflected and a fraction is transmitted. Similarly to Section 3.2, we write thereflected wave4 as

Er = –rE0e–ik0z x (3.139)

Hr = rE0

Z0e–ik0z y (3.140)

where r is the reflection coefficient. The transmitted wave is given by

Et = tE0eikz x (3.141)

Ht = tE0

Zeikz y (3.142)

where t is the transmission coefficient. In the material to the right, the propagationparameter k is given by Equation 3.39 and rewritten here for convenience,

k =2πλ0

[ε′ + i

(ε′′ +

σ

ωε0

)]1/2

At z = 0 the total parallel field amplitudes on either side of the boundary must becontinuous across it. Therefore:

Ei + Er = Et (3.143)

Ei – rEi = tEi (3.144)

Hi + rHi = tHi (3.145)

EiZ0

+ rEiZ0

= tEiZ

(3.146)

4 The choice of phase for the reflected wave is a matter of convention. The reflected waves could also bewritten Er = rE0e–ik0zx and Hr = –r(E0/Z0)e–k0zy. The first convention, Equations 3.139 and 3.140, is oftenused in physical optics while the second is common in electrical engineering. The second choice results in achange of sign for the reflection coefficient with the consequence that r = (Z – Z0)/(Z0 + Z) and 1 = t – r.


From Equations 3.144 and 3.146 we have

1 – r = t (3.147)1Z0

+rZ0

=tZ

(3.148)

From these two relations we find:

1 = r + t (3.149)

r =Z0 – ZZ0 + Z

(3.150)

t =2Z

Z0 + Z(3.151)

3.2.3.2 Lossless dielectric

For a lossless dielectric ε′′ = 0 and σ = 0 so the propagation vector becomes k =(2π /λ0) ε′. The Poynting vector on the left-hand side of the boundary is

Sz<0 = (Ei + Er)×(H∗i +H∗r

)(3.152)

=(E0eik0x – rE0e–ik0x

)× (E0

Z0e–ik0z + r

E0

Z0eik0z

)z

=E20

Z0

[(1 – r2

)– i2r sin (2k0z)

]z (3.153)

At z = 0 the energy flux from the left is

S– =E20

Z0

(1 – r2

)z (3.154)

and since r < 1, the net energy flux propagates from left to right. In the region z < 0 theoscillatory term in Equation 3.153 is due to the standing wave set up by reflection of theplane wave at the boundary. Notice that the phase of the standing wave is in quadraturewith respect to the incident wave. The Poynting vector on the right-hand side is

Sz>0 = (Et)×(H∗t)z (3.155)

= t2E20

Zz (3.156)

The energy flux across the boundary should be continuous, and by using the expressionsfor r and t in terms of the impedances, Equations 3.147 and 3.148, the continuity canbe verified.


3.2.3.3 Lossy dielectric

In the case of a lossy dielectric the propagation vector becomes

k =2πλ0

(ε′ + iε′′

)1/2(3.157)

=2πλ0

(η + iκ) (3.158)

and the wave impedance becomes complex:

Z =

√μ0

ε=√

μ0

ε0 (ε′ + iε′′)=

Z0

η + iκ(3.159)

Therefore, from Equations 3.150 and 3.151 the reflection and transmission coefficientsbecome complex, and the parallel field component continuity conditions at the boundarynow give

t =2

(η + 1) + iκ(3.160)

r =(η – 1) + iκ(η + 1) + iκ

(3.161)

Figure 3.5 shows a plane wave incident from the left on a slightly absorbing slab ofSiO. The shortened wave length and slight loss of amplitude due to absorption can bediscerned within the material. The Poynting vector in the z < 0 region is

Sz<0 =|E0|2Z0

(1 + r∗ei2k0z – re–i2k0z – |r|2) z (3.162)

and in the z > 0 region

Sz>0 = |t|2 |E0|2Z∗

e–2κz z =|E0|2Z0· 4(η – iκ)(η + 1)2 + κ2

e–2κz z (3.163)

At the boundary (z = 0) the flux from the z < 0 region is

S–z=0 =

|E0|2Z0

(1 + r∗ – r – |r|2) = |E0|2

Z0· 4(η – iκ)

(η + 1)2 + κ2(3.164)


1.0

0.5Ei Et

HT

Hr

Ht

Hi

ErET

Air

Air

–800

0

Ele

ctri

c fie

ld (

a.u.

)

–0.5x

Dielectric

Dielectric

–1.0

1.0

0.5

0

Mag

netic

fiel

d (a

.u.)

–0.5

–1.0

–600 –400 –200 0

z (nm)

100 200 300 400 500 600

–800 –600 –400 –200 0

z (nm)

100 200 300 400 500 600

y

z

Figure 3.5 Top panel shows the real part of the incident E-field Ei, reflected E-field Er and totalE-field ET in air to the left of the air-dielectric interface. The wave length of the propagating wavein air is λ0 = 620 nm. Within the dielectric the transmitted E-field Et is shown. The dielectric isSiO with index of refraction n = η + iκ = 1.969 + i0.01175. Bottom panel shows the real part ofthe incident H-field Hi, reflected H-field Hr, and total H-field HT in air to the left of theair-dielectric interface. Within the dielectric the transmitted H-field Ht is shown. Note theshortening of the wave length and the slight decrease in amplitude as the wave penetrates into thelossy dielectric.


and from the z > 0 region

S+z=0 =

|E0|2Z0· 4(η – iκ)(η + 1)2 + κ2

(3.165)

So we see that the complex Poynting vector is conserved across the boundary. When thePoynting vector is complex, we interpret the energy flux or the power flow as the cycle-averaged real part of the Poynting vector. The net energy flux traversing the boundaryfrom the z < 0 region is therefore:

P– =12Re[S–] =

|E0|2Z0· 2η(η + 1)2 + κ2

(3.166)

and the energy flux penetrating the lossy dielectric on the z > 0 side of the boundary is

P+ =12Re[S+] =

|E0|2Z0· 2η(η + 1)2 + κ2

e–2κz (3.167)

Note that the incident energy flux on the boundary from the left is

Pi =12Re[E0eik0z · E

∗0

Z0e–ik0z

]=|E0|22Z0

(3.168)

and the reflected flux is

Pr =12Re[–rE0e–ik0z · r

∗E∗0Z0

eik0z]= –|r|2 |E0|2

2Z0(3.169)

The sum of the incident and reflected power

PT = Pi + Pr =|E0|22Z0

(1 – |r|2) = |E0|2

Z0· 2η(η + 1)2 + κ2

(3.170)

accords with the net power flow calculated from the superposed incident and reflectedfields (Equations 3.162 and 3.166).

3.2.3.4 Good conductor (low frequency regime)

A plane wave incident on a perfect conductor and on a good conductor are shown in Fig-ures 3.6 and 3.7. The difference between ‘perfect’ and ‘good’ is essentially the magnitudeof the conductivity, σ . When σ → ∞ a conductor approaches the perfect conductoridealisation. In the case of a good conductor

k =2πλ0

(ε′ + i

σ

ωε0

)1/2

(3.171)


1.0

0.5

Ei Er

Et = 0

Ht = 0

kr

HT

Hi, Hr

kr

ki

ki

ET

Air

Air

–800

0

Ele

ctri

c fie

ld (

a.u.

)

–0.5

x

Perfectmetal

Perfectmetal

–1.0

2.0

1.0

0

Mag

netic

fiel

d (a

.u.)

–1.0

–2.0

–600 –500–700 –400 –300 –200 –100 0 100 200

y

zz (nm)

–800 –600 –500–700 –400 –300 –200 –100 0 100 200

z (nm)

Figure 3.6 Top panel shows the real part of the incident E-field Ei, reflected E-field Er, and totalE-field ET in air to the left of the air-perfect metal interface. The wave length of the propagatingwave in air is λ0 = 632 nm. Within the perfect conductor (σ →∞) no electromagnetic fieldpenetrates. Bottom panel shows the real part of the incident H-field Hi, reflected H-field Hr, andtotal H-field HT in air to the left of the air-dielectric interface. Within the perfect conductor thetransmitted H-field Ht is null.


1.0

0.5

Ei Er

Et

Ht

Hi

HT

Hr

ETAir

Air

–800

0

Ele

ctri

c fie

ld (

a.u.

)

–0.5

x

Realmetal

Realmetal

–1.0

2.0

1.0

0

Mag

netic

fiel

d (a

.u.)

–1.0

–2.0

–600 –500–700 –400 –300 –200 –100 0 100 200

y

zz (nm)

–800 –600 –500–700 –400 –300 –200 –100 0 100 200

z (nm)

Figure 3.7 Top panel shows the real part of the incident E-field Ei, reflected E-field Er, and totalE-field ET in air to the left of the air-dielectric interface. The wave length of the propagating wavein air is λ0 = 632 nm. Within the dielectric the transmitted E-field Et is shown. Bottom panelshows the real part of the incident H-field Hi, reflected H-field Hr, and total H-field HT in air tothe left of the air-dielectric interface. Within the dielectric the transmitted H-field Ht is shown.

As we saw in the case of silver metal, for good conductors σ / (ωε0) is on the order ofthousands while ε′ is on the order of unity. We can then, to a quite good approximation,write

k � 2πλ0

(iσ

ωε0

)1/2

(3.172)


and recalling from Equation 3.52 that the skin depth is defined as

δ =λ0

2π

(2ωε0σ

)1/2

we can express k as

k �√2iδ

= ±1 + iδ

(3.173)

The sign of k is chosen so that the wave amplitude decays exponentially as it propagatesinto the good conductor in the z > 0 half space. Wave impedance in the conductor is

Z =

√μ0

ε=√μ0

ε0· i–1/2

(ωε0σ

)1/2= Z0 ·

[(1 – i)

(k0δ2

)](3.174)

The reflection and transmission coefficients are

r =2 – (k0δ)2

2 + 2k0δ + (k0δ)2+ i

2k0δ

2 + 2k0δ + (k0δ)2(3.175)

� 1 – ik0δ (3.176)

t =2[(k0δ)2 + k0δ

]2 + 2k0δ + (k0δ)2

– i2k0δ

2 + 2k0δ + (k0δ)2(3.177)

� (1 – i)k0δ (3.178)

At the boundary (z = 0), the flux from the z < 0 region is

S–z=0 =

|E0|2Z0· (1 + r∗ – r – |r|2)

=|E0|2Z0· 4k0δ

2 + 2k0δ + (k0δ)2(1 – i) (3.179)

and the flux penetrating into the conductor (z > 0) is

Sz>0 =|E0|2 |t|2Z∗

e–(2/δ)zz

=|E0|2Z0· 4k0δ

2 + 2k0δ + (k0δ)2(1 – i)e–(2/δ)zz (3.180)


At the z = 0 boundary, the flux from the z > 0 region is

S+z=0 =

|E0|2 |t|2Z∗

=|E0|2Z0· 4k0δ

2 + 2k0δ + (k0δ)2(1 – i) (3.181)

and it is clear that the complex Poynting vectors S–z=0 = S+

z=0. Because the skin depthis only a few nanometres we see that the energy flux penetrates significantly only a fewtens of nanometres. As in the case of the lossy dielectric, the power penetrating into themetal is given by

P+ =12Re[S+z=0

](3.182)

=|E0|2Z0· 2k0δ

2 + 2k0δ + (k0δ)2(3.183)

And the power entering the metal from the z < 0 side is

P– =12Re[S–z=0

](3.184)

=|E0|2Z0· 2k0δ

2 + 2k0δ + (k0δ)2(3.185)

So we see that P– = P+ and the power across the boundary is conserved. Furthermore,we can confirm that the sum of the incident power Pi and reflected power Pr is equal tothe power transmitted across the boundary:

Pi + Pr =12|E0|2Z0

(1 – |R|2) (3.186)

=|E0|2Z0· 2k0δ

2 + k0δ + (k0δ)2(3.187)

Now, from the earlier example in Section 3.1.4.2 for an air-silver metal interface, thevalue for k0 = 1.22 × 107 m–1, the skin depth δ = 2.6 × 10–9 m, and the product k0δ =3.2× 10–2. Therefore:

P– = P+ � |E0|2Z0· k0δ (3.188)


The cycle-averaged incident power is

Pi =|E0|22Z0

(3.189)

and therefore the fraction of the incident power entering the metal is

P+

Pi= 2k0δ = 6.4× 10–2 (3.190)

About 6% of the incident power is transmitted to the metal.The reflected power is given by

Pr = –|r|2 |E0|22Z0

(3.191)

and the fraction of the incident power reflected is

PrPi� (1 – k0δ) = 96.8× 10–2 (3.192)

We see that about 97% of the incident power is reflected.

3.2.3.5 Current density and Joule heating

We can easily calculate the magnitude and phase of the current density induced in thereal metal by using Ohm’s law and the field continuity conditions at the interface. FromOhm’s law we have

J = σEtx (3.193)

and from the continuity condition

Ei(1 – r) = Eit = Et

For a high-conductivity material we have

Z = Z0 · 1√εm

= Z0

√ωε0

iσ= Z0e–iπ /4

√ωε0

σ(3.194)

where the expression for the conductor permittivity εm is obtained from Equation 3.172.Since |Z| |Z0|, the transmission coefficient t can be written as

t =2Z

Z0 + Z� 2ZZ0

= 2 e–iπ /4√ωε0

σ(3.195)


So the transmitted and incident E-field amplitudes are related by

Et = 2 e–iπ /4√ωε0

σE0 (3.196)

We see that the transmitted E-field exhibits a phase lag with respect to the incident E-field by π /4, and Equation 3.193 shows that the current density induced in the metal alsolags the incident E-field by π /4.

Both the current density J and the transmitted E-field are rapidly attenuated exponen-tially as they penetrate the conductor. From Equation 3.173 we can write the propagationparameter in the conductor in terms of the skin depth:

J(z) = 2√σωε0 E0 ei(z/δ–π /4)e–z/δx (3.197)

Et(z) = 2

√ωε0

σE0ei(z/δ–π /4)e–z/δx (3.198)

The power density (Wm–3) associated with ‘Joule heating’ in the conductor is given by

dPJoule

dV= Et · J∗ (3.199)

In order to compare this Joule heating power to P+ or P– we need first to integrate overthe penetration depth in the z direction, then integrate over a cross-sectional area ofone square metre in the x – y plane in order to calculate the volume power dissipated inthe conductor. This power dissipation corresponds to the third term on the right in thepower-balance, Equation 3.65:

PJoule =12Re[∫ 1

0

∫ 1

0

∫ ∞0

Et · J∗ dxdydz]

= 2ωε0 |E0|2 δ2

=|E0|2Z0· k0δ (3.200)

and we see that this result agrees with our earlier power calculation, Equation 3.188,based on the Poynting vector energy flux entering the conductor normal to its surface.

3.2.4 Plane waves in a lossy, conductive medium (highfrequency limit)

In Sections 3.1.4 and 3.2.3 we developed the physics of a plane electromagnetic waveinteracting at the surface of, and within, a good conductor (most often encountered


as a real metal). This development assumed that the conductivity σ is real. We showhere that, although this assumption is adequate for radio and microwave frequencies,the conductivity must be considered complex in the optical range of the electromagneticspectrum.

We start with the wave equation for the E-field component of an electromagnetic wavepropagating in a medium, Equation 3.36, that we rewrite here for convenience:

∂2Ex∂z2

+ μεω2(1 + i

σ

ωε

)Ex = 0 (3.201)

and assume, as usual, a plane wave solution as in Equation 3.29:

Ex = E0ei(kzz–ωt) (3.202)

Substituting Equation 3.202 into Equation 3.201 results in

k2z = μεω2 + iμσω (3.203)

As before, we assume non-magnetic materials (μ = μ0) and consider the permittivity tobe complex, ε = ε0 (ε′ + iε′′). But now we consider the conductivity to be complex aswell, σ = σ ′ + iσ ′′. Now we rewrite kz as km to emphasise wave propagation inside themetallic conductor; and recognise that it is complex as well:

km = k′m + ik′′m (3.204)

Substituting the complex quantities into Equation 3.203 and gathering the real andimaginary parts,

k′ 2m – k′′ 2m = μ0ε0ε′ω2 – μ0σ

′′ω

= μ0ε0ω2[ε′ –

σ ′′

ε0ω

]= k20

[ε′ –

σ ′′

ωε0

]= α2 (3.205)

2k′mk′′m = μ0ε0ε

′′ω2 + μ0σ′ω

= μ0ε0ω2[ε′′ +

σ ′

ε0ω

]= k20

[ε′′ +

σ ′

ωε0

]= β2 (3.206)

The two coupled equations, Equations 3.205 and 3.206, can be solved for the real andimaginary parts of km:

k′m = ± α√2

√1±

√1 + (β/α)4 (3.207)

k′′m = ± α√2

√–1±

√1 + (β/α)4 (3.208)


We also know that the propagation parameter within the metal km is related to the free-space propagation parameter k0 through the complex index of refraction n in the metal:

(kmk0

)2

= n2 = (η + iκ)2 (3.209)

But n is also related to the real and complex ‘effective’ dielectric constants in the metalby

n2 = ε′eff + iε′′eff (3.210)

The relations between η, κ and ε′eff, ε′′eff are

η =

√ε′eff2

√1±

√1 +

(ε′′eff/ε

′eff

)(3.211)

κ =

√ε′eff2

√1±

√–1 +

(ε′′eff/ε

′eff

)(3.212)

But:

k′m = k0η (3.213)

k′′m = k0κ (3.214)

Then comparing Equations 3.213 and 3.214 to Equations 3.207 and 3.208 we see that

ε′eff = ε′ –

σ ′′

ωε0(3.215)

ε′′eff = ε′′ +

σ ′

ωε0(3.216)

The two terms ε′, ε′′ are the real and imaginary parts of the metal dielectric constantexcluding the highly polarisable conduction electrons. They represent the response ofthe much more tightly bound valence electrons to the transmitted plane wave. Theterms involving σ ′, σ ′′ represent the frequency-dependent response of the conductionelectrons to the transmitted wave. In high-conductivity materials these latter two termsdominate the response. Now real metals are dispersive which means that ε′eff and ε

′′eff

depend on the frequency. A simple dispersion relation can be found if the metal con-ductor is modelled as a harmonically oscillating free-electron gas. The harmonic motionis driven at the incident wave frequency ω and damped at a rate �, inserted into themodel phenomenologically and corresponding to acoustic and radiative dissipation.

3.2.4.1 Damped harmonic oscillator model for conduction current

We posit that the equation of motion of the conduction electrons is governed by har-monic acceleration driven by a plane-wave electromagnetic field propagating in the metal


and polarised along the x direction:

med2xdt2

+me�dxdt

= eEt = eTE0eikmz–ωt (3.217)

where me, e are the electron mass and charge, � a phenomenological damping constant,and eEt = eTE0ei(kmz–ωt) the driving force on the conduction current. Dropping thecommon e–iωt factor, we have for the solutions of position, velocity, and acceleration:

x = –1

me(ω2 + i�ω

) eTE0 (3.218)

dxdt

= iω

me(ω2 + i�ω

) eTE0 (3.219)

d2xdt2

=ω2

me(ω2 + i�ω

) eTE0 (3.220)

Now the amplitude of the conduction current density is given by

Jc = eNedxdt

= eNeω

me

(�ω – iω2) eTE0 (3.221)

where the electron density in the metal conduction band Ne is related to the resonancefrequency of the oscillating electrons ωp by

ω2p =

e2Ne

meε0(3.222)

The resonant frequency is called the ‘bulk plasmon frequency’. The conduction currentJc can now be written as

Jc =ω2p

� – iωε0TE0 (3.223)

But from the standard constitutive relation we have

Jcx = σEtx =1ε0σε0TE0x (3.224)

Comparing Equations 3.224 and 3.223, and separating real and imaginary parts, we seethat

σ

ε0=

�(�2

ω2p+ ω2

ω2p

) + iω(

�2

ω2p+ ω2

ω2p

) (3.225)


Now from Equation 3.222 we can estimate the bulk plasmon frequency and compareit to ω and �. A high-conductivity metal such as silver exhibits a conduction electrondensity Ne � 6×1028 m–3 and therefore ωp � 1.3×1016 s–1. The damping rate � due toradiative loss and acoustic coupling is typically ∼ 1014 s–1, and ω in the visible and nearIR is ∼ 1015 s–1. Therefore, we can write

σ

ε0� �ω

2p

ω2+ iω2p

ω(3.226)

Thus, at relatively low frequencies (radio and microwave) the real term dominates, andthe conductivity is said to be ‘ohmic’. As the driving frequency into the optical range,however, the imaginary term dominates. At ω ∼ 1015 s–1 the real term is only about5% of the imaginary term. We can, from Equations 3.226, 3.215, and 3.216, write thefrequency dependence of the ‘effective’ dielectric constants in the metal as

ε′eff =

(ε′ –

ω2p

ω2

)(3.227)

ε′′eff =

(ε′′ +

�ω2p

ω3

)(3.228)

The valence electrons in a metal are very tightly bound with a large energy gap be-tween the valence ground and excited states. Therefore, we can to good approximationwrite ε′ = 1 and ε′′ = 0. The effective, frequency-dependent dielectric constants thensimplify to

ε′eff �(1 –

ω2p

ω2

)(3.229)

ε′′eff �(�ω2

p

ω3

)(3.230)

Returning to Equation 3.226 we see that, at high frequency

σ → iσ ′′ = iω2p

ωε0 (3.231)

and the E-field in the conductor becomes

Et = TE0→ 2e–iπ /2√ωε0

σ ′′E0 (3.232)

Further reading 85

The E-field in the conductor now lags the incident driving field by π /2 and theconduction current, from Equation 3.224, becomes

Jc→ iω2p

ωε0Et = i

ω2p

ωε0TE0 = 2

√σ ′′ωε0E0 = 2ωpε0E0 (3.233)

We see that, in the optical frequency regime, the conduction current induced in themetal is in phase with the driving field, proportional to the incident amplitude, andindependent of the driving field frequency.

3.3 Summary

In this chapter we have reviewed the principal physical attributes of plane waves. Westarted with introducing the phasor form of the vector field and then showed how Max-well’s equations could be decoupled to form second-order differential equations of eachvector field E and B. An important class of solutions to these equations take the phasorform. From there we introduced the scalar Helmholtz equations for the spatial depend-ence of the field amplitudes. Then we discussed plane waves in lossy dielectrics and goodconductors in the low frequency limit. Energy densities and energy flux of plane waveswere then presented, taking into account dissipative loss. The standard Fresnel relationsreflection and transmission of amplitude and power were then discussed and applied tototal internal reflection, reflection, and transmission at a material interface under lossyconditions. Expressions for plane waves travelling in a lossy conductive medium (realmetals) at both a low and high frequency limit were then developed and applied to thedamped harmonic oscillator model for the conduction current.

3.4 Further reading

1. M. Born and E Wolf, Principles of Optics, 6th edition, Pergamon Press (1980).



4. J.-P. Pérez, R. Carles, and R. Fleckinger, Électromagnétism Fondements et applications,3ème édition, Masson (1997).

5. H. A. Haus,Waves and Fields in Optoelectronics, Prentice Hall (1984).


4

Energy Flow in Polarisable Matter

4.1 Poynting’s theorem in polarisable material

Recall from Chapter 2, Equation 2.60 that the energy flow across the surface of a spatialvolume is

dEmech

dt+dEem

dt= –

∮E×H · da (4.1)

and that the integrand on the right-hand side is identified with the energy flux crossingthe surface,

S = E×H (4.2)

with S called the Poynting vector. The two terms on the left are the rate of change of‘mechanical’ energy stored in any material included in the bounded volume and the rateof change of ‘field’ energy associated with the E- and B-fields within the same volume.Let us deconstruct some of the characteristics of the Poynting vector to bring out theexplicit time-dependent terms. Firstly,

∇ · S = ∇ · (E×H) (4.3)

= –E · (∇ ×H) +H · (∇ × E) (4.4)

From the Maxwell–Faraday law and the Maxwell–Ampère law, Equations 2.27 and 2.28,we have

∇ × E = –∂B∂t

= –μ0∂H∂t

(4.5)

and

∇ ×H =∂D∂t

+ Jfree (4.6)


Harmonically driven polarisation field 87

so that

∇ · S = – E ·(∂D∂t

+ Jfree

)+H ·

(–μ0

∂H∂t

)(4.7)

= –(E · Jfree + E · ∂D

∂t+ μ0H · ∂H

∂t

)(4.8)

Now we specify the case where there are no free currents (Jfree = 0), but the materialwithin the enclosed volume is polarisable. Then the displacement field is

D = ε0E + P

and

–∇ · S =(E · ∂D

∂t+ μ0H · ∂H

∂t

)(4.9)

=[12ε0 · ∂

∂t(E · E) + 1

2μ0∂

∂t(H ·H)

]+ E · ∂P

∂t

=∂

∂t

[12

(ε0E2 + μ0H2)] + E · ∂P

∂t(4.10)

The term in brackets on the right-hand side of Equation 4.10 is the field energy andthe second term is the time rate of change of the mechanical energy of the material. Wesee that the flow of energy into or out of any material included within the volume iscontrolled by the time dependence of the polarisation field in that material.

4.2 Harmonically driven polarisation field

The polarisation can be considered a density of harmonic dipole oscillators characterisedby a ‘natural’ frequency ω0 and a damping frequency γ . If a harmonically oscillatingexternal electric field E is applied to this dipole density, the equation of motion of theresponse will be

∂2P∂t2

+ γ∂P∂t

+ ω20P = ω2

dε0E (4.11)

where ω2d is a ‘strength of external field coupling’ parameter and E is the external driving

field. The coupling parameter ωd clearly is the frequency of the driving field E. After afew lines of tedious but straight-forward algebra we find that

88 Energy Flow in Polarisable Matter

E · ∂P∂t

=

stored︷︸︸︷1

ε0ω2d

12∂

∂t

⎡⎢⎢⎢⎣(∂P∂t

)2

︸︷︷︸kinetic

+ ω20P

2︸︷︷︸potential

⎤⎥⎥⎥⎦+

dissipative︷︸︸︷γ

ε0ω2d

(∂P∂t

)2

(4.12)

Remembering that for a polarisable but non-magnetic material the bound currentdensity is given by,

Jbound =∂P∂t

(4.13)

and

dEmech

dt= E · Jbound (4.14)

we see immediately that Equation 4.12 represents the flow of mechanical energy intoand out of the ensemble of electric dipoles constituting the material of the system. Threeterms constitute this mechanical energy: the kinetic and potential energy stored in thedipoles, and the dissipated mechanical energy associated with the damping rate γ asdenoted in Equation 4.12.

4.3 Drude–Lorentz dispersion

Equation 4.12 describes the rate of change of mechanical energy in terms of the timedependence of the polarisation field, but the response of matter to an external driv-ing field varies with the frequency. If the response to the driving field is linear, thepolarisation field is proportional to the applied electric field.

P = ε0χE (4.15)

where χ is the linear susceptibility of the material. Strictly speaking, the susceptibilityassociated with P should be written χe to distinguish it from the magnetic suscepti-bility χm, but here we drop the distinction since it is obvious from the context. Thesusceptibility is frequency dependent and can be complex, χ(ω) = χ ′(ω) + iχ ′′(ω). Therelation characterising the material frequency response is called the material dispersionand is usually expressed as the frequency dependence of the complex dielectric con-stant ε(ω) = ε′(ω) + iε′′(ω). For linear materials the relation between susceptibility anddielectric constant is given by

χ(ω) = ε(ω) – 1 (4.16)

A simple dispersion relation for metals is the Drude–Lorentz model that expresses thefrequency dependence of the dielectric constant:

Drude–Lorentz dispersion 89

ε(ω) = 1 +ω2p

ω20 – ω

2 – iγω(4.17)

In Equation 4.17, ω is the driving frequency, ω0 is the resonance frequency, γ is theirrecoverable rate of energy dissipation to ‘heat’, and ωp is the bulk plasmon frequencyof the electrons in the conduction band of the metal. Separating the real and imaginaryparts of ε,

ε(ω) =

[1 +

ω2p

(ω20 – ω

2)

(ω20 – ω

2)2 + (γω)2

]+ i

[γω2

pω(ω20 – ω

2)2 + (γω)2

](4.18)

= ε′ + iε′′

Now, from the constitutive relation between the displacement field and the electric field,and the definition of the displacement field, we can write

D = ε0εE (4.19)

= ε0E + P (4.20)

So we can write the polarization field in terms of the dielectric constant and the electricfield as

P = (ε – 1)ε0E (4.21)

Therefore, we can express the factor in square brackets in the stored energy part ofEquation 4.12:

(∂P∂t

)2

+ ω20P

2 = (ε – 1)2ε20

[(∂E∂t

)2

+ ω20E

2

](4.22)

The right-hand side of Equation 4.22 is complex due to the (ε – 1)2 factor. In orderto develop expressions for the stored and dissipated energies, it is therefore convenientto express the fields in complex representation and calculate the absolute value of theenergy terms. In complex notation the E-field and its time derivative are written as

E = E0e–iωt and∂E∂t

= –iωE

and the absolute value of the stored energy factor is∣∣∣∣∣(∂P∂t

)2∣∣∣∣∣ + ω2

0

∣∣P2∣∣ = ∣∣(ε – 1)2∣∣ ε20

[∣∣∣∣∣(∂E∂t

)2∣∣∣∣∣ + ω2

0

∣∣E2∣∣]

(∣∣∣∣∂P∂t∣∣∣∣)2

+ ω20|P|2 =ε20

(ω20 + ω

2) [(ε′ – 1)2 + (ε′′)2]E20 (4.23)


Now from the dispersion relation Equation 4.17 we can also write

(ε′ – 1

)2+(ε′′)2

=ω4p

(ω20 – ω

2)2[(

ω20 – ω

2)2

+ (γω)2]2 +

γ 2ω4pω

2[(ω20 – ω

2)2

+ (γω)2]2 (4.24)

=ω4p(

ω20 – ω

2)2 + (γω)2

(4.25)

Therefore, from Equations 4.22, 4.23, and 4.25 we have the absolute value of the storedenergy from Equation 4.12

12ε0ω2

p

[(∣∣∣∣∂P∂t∣∣∣∣)2

+ ω20|P|2

]=ε0

2

[ω2p

(ω20 + ω

2)

(ω20 – ω

2)2

+ (γω)2

]E20 (4.26)

By ‘stored energy’ we mean the part of the energy density that flows from the fields intothe dipoles, is stored there, and can flow back to the fields.

Making analogous substitutions we have for the dissipative part of dEmech/dt,

γ

ε0ω2p

(∂|P|∂t

)2

=γω2

pω2(

ω20 – ω

2)2

+ (γω)2ε0E2

0

= ε′′ε0ωE20 (4.27)

This dissipative loss is irreversible and cannot be recovered by the fields.Returning to the absolute value of the stored energy, the factor in square brackets on

the right-hand side of Equation 4.26 can be written as the sum of two fractions,

ω2p(ω

2 + ω20)(

ω20 – ω

2)2

+ (γω)2=

ω2p

(ω20 – ω

2)

(ω20 – ω

2)2

+ (γω)2+

2ω2ω2p(

ω20 – ω

2)2

+ (γω)2

and from Equation 4.18 we can identify the first term as

ε′ – 1 and the second as2ωγε′′

Therefore:

12ε0ω2

p

[(∂|P|∂t

)2

+ ω20|P|2

]=ε0

2

[(ε′ – 1) +

2ωγε′′]E20 (4.28)


4.3.1 Stored energy density

From Equations 4.10 and 4.12, and Stokes’ theorem, we can write the energy conserva-tion theorem as

–∫AS · da =

∫V

∂

∂t

{12

[ε0E2 + μ0H2+

1ε0ω2

p

((∂P∂t

)2

+ ω20P

2

)]}dV+

∫V

γ

ε0ω2p

(∂P∂t

)2

dV (4.29)

Then taking the absolute value of the right-hand side of Equation 4.29, and substitutingEquation 4.27 and Equation 4.28 into it, we have

–∫AS · da =

∫V

∂

∂t

{12

[ε0

(ε′ +

2ωγε′′)E2+μ0H2

]}dV+

∫Vε0ε′′ωE2dV (4.30)

and writing the magnetic field energy in terms of the electric field energy

H2 =ε0

μ0|ε|E2 =

ε0

μ0

√ε′2 + ε′′2E2 (4.31)

the stored energy density Us within the curly brackets on the right-hand side ofEquation 4.30 can be written

Us =ε0

2

[(ε′ +

2ωγε′′)+√ε′2 + ε′′2

]E2 (4.32)

and averaging Us over an optical cycle

〈Us〉 = ε0

4

[(ε′ +

2ωγε′′)+√ε′2 + ε′′2

]E20 (4.33)

In the limit of a lossless material, ε′′, γ → 0,

〈Us〉 = 12ε0ε′E2

0 (4.34)

as expected.


Finally, we can express this conservative energy density in terms of the real andimaginary parts of the index of refraction, η and κ, respectively:

ε′ =(η2 – κ2

)(4.35)

ε′′ = 2ηκ (4.36)

So that

Us = ε0

(η2 +

2ωηκγ

)E2 (4.37)

and averaging Us over an optical cycle,

〈Us〉 = ε0

2

(η2 +

2ωηκγ

)E20 (4.38)

4.3.2 Dissipated energy density

The rate of energy density dissipation ∂Ud /∂t is given by the integrand of the secondterm on the right-hand side of Equation 4.30:

∂Ud

∂t=

γ

ε0ω2p

(∂P∂t

)2

(4.39)

Remembering that P = ε0(ε – 1)E, we have

P2 = ε20[(ε′ – 1

)2+(ε′′)2]

E2 (4.40)

As usual, we take E to be a harmonically oscillating field, E = E0e–iωt. Then:

P = ε0

√(ε′ – 1)2 + (ε′′)2E (4.41)

and

∂P∂t

= ε0

√(ε′ – 1)2 + (ε′′)2

∂E∂t

(4.42)

So, substituting Equation 4.42 into Equation 4.39, we have

∂Ud

∂t=γ ε0ω

2

ω2p

[(ε′ – 1

)2+(ε′′)2]

E2 (4.43)


where E0 is the electric field amplitude. This expression can be further simplified bynoting that

(ε′ – 1

)2 + (ε′′)2 = ω4p(

ω20 – ω

2)2

+ (γω)2

and therefore, taking into account Equation 4.27,

∂Ud

∂t= ε0ε′′ωE2 (4.44)

Averaging over an optical cycle, ⟨∂Ud

∂t

⟩=

12ε0ε′′ωE2

0 (4.45)

and the average energy dissipated in one optical cycle (T = 2π /ω) is

〈Ud〉 = πε0ε′′E20 (4.46)

4.3.3 Time dependence of stored and dissipated energy density

Representing the driving E-field as a real quantity, E = E0 cosωt, we see fromEquation 4.37 that the stored energy is

Us = ε0

(η2 +

2ωηκγ

)E20 cos

2 ωt (4.47)

and is therefore in phase with the driving field energy flux, Sz = ExHy cos2 ωt. Therefore,the rate of change of stored energy is in quadrature:

dUs

dt= –2ωε0

(η2 +

2ωηκγ

)2

E20 cosωt sinωt (4.48)

In contrast, from Equation 4.44, the rate of change of the dissipated energy density is inphase with the driving field energy flux

dUd

dt= ε0ε′′ωE2

0 cos2 ωt (4.49)

while the dissipated energy itself accumulates over the optical cycles

Ud = ε0ε′′E20

(ωt2

+sin 2ωt

4

)(4.50)


50

t (ps)

40

30

20

10

00 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

t (ps)

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

2

3.0

2.4

1.8

1.2

0.6

0

4.0

3.2

2.4

1.6

0.8

0

1

0

–1

–2

2πω

2πω

d

(x1017)

US

ε0E02dt

d

(x1015)

US

ε0E02dt

US

ε0E02

Ud

ε0E02

Figure 4.1 Top panel: The full curve shows time dependence of stored energy (normalisedto ε0E

20 ), Equation 4.47, and the dashed curve shows time dependence of the rate of

change of stored energy, Equation 4.48. Bottom panel: The full curve shows accumulationof dissipated energy with time, Equation 4.50 and the dashed curve shows rate of changeof dissipated energy (dissipated power), Equation 4.49. These graphs are plotted with thefollowing parameters taken for silver metal: ε′ = –9.108, ε′′ = 0.753,ω = 3.768× 1015 s–1,(λ = 500 nm), γ = 1.0× 1014 s–1.

Figure 4.1 shows the time dependence of the various constituents of the energy densityin a polarisable medium.

4.3.4 Frequency dependence of stored and dissipated energy

Returning to the expression for the stored energy in terms of the real and imaginaryparts of the dielectric constants, Equation 4.32, and taking the limit of negligible loss,

Us � ε0ε′E2 (4.51)

The dispersion relation of the Drude-Lorentz model for ε′ is

Polarisation from polarisability 95

ε′(ω) =ω2p

(ω20 – ω

2)

(ω20 – ω

2)2

+ (γω)2+ 1 (4.52)

and therefore the frequency dependence of the stored energy in lossless media is

Us(ω) � ε0[

ω2p

(ω20 – ω

2)

(ω20 – ω

2)2 + (γω)2

+ 1

]E2 (4.53)

The stored energy exhibits the same ‘dispersive’ frequency dependence as a driven har-monic oscillator, in phase with the driving field below resonance, ω0 – ω > 0 and π outof phase at frequencies above resonance, ω0 – ω < 0. In the simplest case of a metalmodelled with a single resonance at the bulk plasmon frequency, ω0 = ωp, the storedenergy responds in phase at all frequencies ‘to the red’ of ωp. For most common noblemetals ωp is in the blue or near ultraviolet regions of the spectrum. Therefore, for inci-dent driving fields in the visible or near-infrared, the usual case in plasmonic studies, thestored energy is in phase with the incident energy flux.

The dispersion relation for the material loss term is

ε′′(ω) =γωω2

p(w20 – ω

2)2

+ (γω)2(4.54)

and, using Equation 4.50, the dissipative energy density is peaked at the resonance fre-quency ω0, decreasing symmetrically as the driving frequency is tuned above or belowresonance.

Ud = ε0

[γωω2

p(ω20 – ω

2)2

+ (γω)2

]E20

(ωt2

+sin 2ωt

4

)(4.55)

The dispersive and absorptive frequency profiles are simply those of a driven, dampedharmonic oscillator. This behaviour is consistent with the polarisation of the materialmodelled as a density of dipoles.

4.4 Polarisation from polarisability

In Chapter 2 the polarisation vector field was introduced in Equations 2.15 and 2.17.The polarisation field P adds vectorially to the applied electric field (multiplied by thepermittivity) ε0E to produce the total displacement field D, and in many cases, thepolarisation field itself is proportional to the applied electric field. The proportional-ity constant, the susceptibility χ , is a property of the material; and the dielectric constantof the material is simply related to this susceptibility by ε = 1 + χ . As the term suggests,the susceptibility is a measure of the response of the material to an external applied field.Since matter is actually composed of atoms and molecules, the macroscopic response


must somehow be related to the distortion of local electric fields around the constituentmicroscopic particles. The goal of this section is to develop that relation.

4.4.1 Electric field inside a material

At the truly microscopic distance scale, comparable to the distance between atoms incondensed matter, the electric field due to electronic and nuclear charge densities isextremely complicated and constantly changing in time due to the orbital motion ofelectron charge densities around nuclei and the thermal motion of the nuclei themselves.The calculation of this field is hopeless, but in fact, it is not the field we really seekanyway. What we really want is the field at a fixed point inside the material due to twoinfluences: the external E-field applied to the bulk material and the E-field internal to thematerial arising from its polarisation. Figure 4.2 is a sketch of the various elements of theproblem. The figure shows a slab of material with susceptibility χ , dielectric constant ε,placed between two conducting plates across which an electric field Eext is applied. Thefield at a point internal to the material can be considered the sum of two superposedfields. The source of the first field is the charge density induced at the surface of thematerial by the applied external field. This surface charge density produces a uniformpolarisation field P opposite in direction to the applied E-field. The E-field due to thispolarisation field is Eint = –P/ε0. The source of the second field are the microscopicinduced dipoles themselves. As we have already discussed above, at a spatial resolution

+

Eext

Eslab

Pslabε0

= –

X

Z

+

+

+

+

–

–

–

–

–

–

–

–

–

–

+

+++

+

+++

++ + –

––

––

–θ

––

– + –

+ –

+ –

+–

+–

+–

+

+

+σP

slabσPslab

σPsphere

Figure 4.2 External electric field Eext is applied to two externalconducting parallel plates. The material slab with dielectricconstant ε and susceptibility χ exhibits a polarisation field –P thesource of which is the surface charge density σP induced at theslab-plate interface. The volume of the sphere inside the slab islarge with respect to the size of individual dipoles comprising thematerial but small compared to the bulk volume of the slab itself.

Polarisation from polarisability 97

of an individual dipole, the E-field is neither practically calculable nor useful. We seekinstead the field at a specific point due to the average influence of many induced dipoles.To this end we construct a sphere whose volume is large with respect to the volume ofan individual dipole but negligible compared to the bulk volume. Using Gauss’s law, wecan compute this E-field EP at the centre of the sphere due to the bound charge densityaround the surface. The net field at the centre of the sphere, experienced by a moleculeat that site, is then Eext – P/ε0 + EP .

The displacement field D = ε0Eext is perpendicular to the interface and thereforemust be continuous across the slab boundary, and therefore, the E-field at the point ofinterest due to the external applied field is

Dε0

= Eext +Pε0

(4.56)

The surface charge density at the two slab interfaces σ slabP is the source of a polarisationfield at all points within the slab equal to –P/ε0. Now we invoke Gauss’s law to calculatethe E-field EP at centre of the sphere from the polarisation charge density σ sphereP at thesphere surface. This bound charge density is determined from the polarisation compo-nent normal to the sphere, and therefore the distribution of charge density around thesurface goes as σ sphereP (θ) = |P| cos θ where θ is the angle between the z-axis and thesurface normal as shown in Figure 4.2. The E-field at the sphere centre is then

EP =1

4πε0

∫S

σsphereP cos θ

r20dS (4.57)

where r0 is the sphere radius. The surface differential is dS = r20 sin θ dθ dϕ, andintegrating over the azimuthal angle ϕ we have

dS = 2πr20 sin θ dθ (4.58)

Putting σ sphereP (θ) and dS into Equation 4.57 we have

EP =|P|2πr204πε0r20

∫ π

0sin θ cos2 θ dθ (4.59)

=|P|3ε0

(4.60)

and

EP =P3ε0

(4.61)

The effective E-field at the centre of the sphere, experienced by a material moleculeplaced at that position, is therefore


Eeff =Dε0

–Pε0

+ EP (4.62)

= Eext +Pε0

–Pε0

+P3ε0

(4.63)

Eeff = Eext +P3ε0

(4.64)

Note that the effective E-field does not depend on the radius of the sphere. Any spherewill yield the same result as long as the spherical volume is not comparable to an indi-vidual dipole element. The only question that remains is the possible contribution of theE-fields of individual dipole elements inside the sphere. Since the material was initiallyassumed isotropic with no net polarisation in the absence of an external field, we cansafely assume that there is no E-field contribution to the dipole elements themselves.

4.4.2 Polarisation and polarisability

Now we want to use Eeff to obtain the macroscopic polarisation field from the micro-scopic dipole moment induced in each molecule. The connection from the micro- tothe macro-world is through the polarisability, a property of matter at the molecular level.The polarisability α is defined through the relation,

p = αε0Eeff (4.65)

where p is the induced dipole moment of an individual molecule. Since the macroscopicpolarisation P can be interpreted as a dipole density, we can write

P = Np = Nαε0Eeff = Nαε0

(E +

P3ε0

)(4.66)

where N is the number of molecules per unit volume. Remembering that for linearresponse

P = ε0(ε – 1)E (4.67)

We can set Equation 4.66 equal to Equation 4.67 and find

ε – 1ε + 2

=Nα3

=N0 ρm α

3M(4.68)

where in the last term on the right, N0 is Avogadro’s constant, ρm is the density, andM is the molecular mass. This expression is called the Clausius–Mossotti relation. Ittells us how we can calculate the dielectric constant ε, a macroscopic property, from themicroscopic polarisability, α. As such, it furnishes a key bridge between microscopic andmacroscopic physics of matter.

Further reading 99

4.5 Summary

In this chapter we have considered energy flow in electrically polarisable matter. In Sec-tion 4.1 we used the Poynting vector and identified the energy flow associated with the‘mechanical’ energy of the material. In Section 4.2 we identified the stored and dissi-pative parts of this energy flow, and in Section 4.3 we introduced the Drude–Lorentzmodel, commonly used to characterise the polarisation dispersion of many commonmaterials. In the rest of the section we considered the time and frequency dependence,and phase relation of the stored and dissipated energy through an optical cycle. Fi-nally, in Section 4.4 we developed the relation between macroscopic polarisation andmicroscopic polarisability to arrive at the very useful Clausius-Mossotti relation.

4.6 Further reading

1. M. Born and E Wolf, Principles of Optics, 6th edition, Pergamon Press (1980).


3. J. S. Stratton, Electromagnetic Theory, McGraw-Hill Book Company (1941).


5. J.-P. Pérez, R. Carles, R. Fleckinger, Électromagnétism Fondements et applications, 3èmeédition, Masson (1997).

5

The Classical Charged Oscillatorand the Dipole Antenna

5.1 Introduction

Active devices involving oscillating charge confined along linear or pyramidal structuresat the nanoscale constitute a major research and development effort. Here we presentthe basic physics and engineering of the oscillating dipole and some real antennas andantenna arrays whose principles are also pertinent to the nanoscale.

5.2 The proto-antenna

The classical charge dipole, p0 = qa, oscillating at frequency ω (the electromagneticfields of which we considered in Section 2.8), emits radiation of wavelength λ = 2πc/ωand can therefore be considered a prototypical or microscopic antenna radiative source.We recall Equations 2.151 and 2.153, the far-field solutions to Maxwell’s equations for asmall dipole, a λ:

Eθ = –p0 sin θ4πε0

ω2

rc2cos

[ω(t –

rc

)]θ (5.1)


4πω2

rccos

[ω(t –

rc

)]ϕ (5.2)

and taking account ofμ0 = 1/ε0c2, we note that the electric andmagnetic field amplitudesare related by B = E/c. We can associate a characteristic current ic of the dipole withlength a oscillating at frequency ω,

ic =ωp0a

(5.3)


The proto-antenna 101

and a characteristic electric field Ec

Ec =(μ0

ε0

)1/2 ica sin θ2λr

(5.4)

where λ is the wavelength of light at frequency ω. We recall from Equation 2.156 thatthe cycle-averaged energy flux emitted by the oscillator is given by

〈S〉 = μ0p20ω4

32π2c

(sin2 θr2

)(5.5)

or in terms of ic and Ec,

〈S〉 = 18

(μ0

ε0

)1/2 ( ica sin θλr

)2

(5.6)

Figure 5.1 shows the distribution of radiated power density from a point dipole. Thecycle-averaged total power radiated over all space is

〈P〉 = 14πε0

⟨ω4p20

⟩3c3

=2π3

(μ0

ε0

)1/2 (aλ

)2 i2c2

(5.7)

where we have used the fact that⟨ω4p20

⟩= ω4p20/2. Now from a unit analysis, it is easy to

show that (μ0

ε0

)1/2

� 377 ohms (5.8)

zPoint Dipole Power Distribution

x

y

Figure 5.1 Angular distribution of power density (power radiated througha unit area) from a radiating point dipole oriented along z.

102 The Classical Charged Oscillator and the Dipole Antenna

and the cycle-averaged power emitted by the oscillator can be set equal to the averagepower required to drive the oscillator

〈P〉 = 12Rci2c (5.9)

where Rc is the characteristic oscillator impedance to radiation,

Rc =2π3

(μ0

ε0

)1/2

=790 ohms

and for a given dipole length and wavelength (a λ), the radiative impedance Rr is

Rr =2π3

(μ0

ε0

)1/2 (aλ

)2(5.10)

Finally, the directional gain of any antenna,G(θ), is defined as the ratio of power emittedin a given direction to the total radiated power. In the case of the dipole proto-antenna,from Equations 5.5 and 5.7,

G(θ) = 4πr2〈S〉〈P〉 =

32sin2 θ (5.11)

5.3 Real antennas

We consider a real, practical antenna consisting of charge oscillating along a straightconductor of finite length, comparable to the wavelength of the emission. We supposethat the electromagnetic field resulting from such an antenna can be analysed as a linearcombination of proto-antennas arranged along the length of the conductor. Figure 5.2shows the antenna aligned along the z-axis. At point M the field is due to the linearsuperposition of contributions from the proto-antennas arranged along z. One of thesepoint dipoles is shown at position P along z. The E-field atM for one proto-antenna is

E =(μ0

ε0

)1/2 sin θ2λPM

i(z)dz e[–iω

(t– PMc

)]θ (5.12)

We seek to write the distance PM in terms of r and then integrate the resulting expressionfor E over the length of the antenna. Invoking the law of cosines we can write

PM =(r2 +OP2 – 2r OP cos θ

)1/2= r

(1 +

(OPr

)2

–2OP cos θ

r

)1/2

(5.13)

Real antennas 103

M

P

X

Z

dz

O

r

r

θ

θ

Figure 5.2 Geometric disposition for deter-mining the E-field of a line antenna of lengthR at point M by integrating contributionsfrom point dipoles aligned along z.

At distances far from the antenna where r � OP we have PM → r. Then substitutingthis result into Equation 5.12, and using k = ω/c, we have

E(M) �(μ0

ε0

)1/2 sin θ2λr

[∫ +R/2

–R/2ic(z)eikzzdz

]e–[ω(t–

rc )]θ (5.14)

where

kz = k cos θ =(2πλ

)

5.3.1 The half-wave antenna

We posit that the antenna length is λ/2 and that a proto-dipole oscillator source, locatedat the midpoint, is driving current to the antenna extremities. A standing wave is set upalong the antenna such that

ic(z) = ic cos(2πzλ

)= ic cos (kz) (5.15)

Then the integral in Equation 5.14 becomes

Ic(k) =∫ λ/4

–λ/4ic cos

(2πzλ

)eik cos θzdz (5.16)

and the resulting E-field at P is

Ehw(P) = –(μ0

ε0

)1/2 ( ic2πr

)cos

[π cos

(θ2

)]sin θ

cos[ω(t –

rc

)]θ (5.17)


The cycle-averaged energy flux for the half-wave dipole antenna, 〈S〉hw, is

〈S〉hw =1

8π2ε0c

(i2cr2

)cos2

[π cos

(θ2

)]sin2 θ

(5.18)

and the total cycle-averaged power of the half-wave antenna is

〈P〉hw =1

4πε0ci2c

∫ π

0

cos2[π cos

(θ2

)]sin θ

dθ (5.19)

〈P〉hw =36.5i2c

〈P〉hw =12i2c Rr

and therefore the radiative impedance of the half-wave antenna is Rr = 73 ohms. Atypical FM radio emission frequency is 100 MHz, and from the above calculation, wesee why a typical FM radio antenna consists of a 75 ohm cable about 1.5 metres inlength.

5.3.2 Array of half-wave antennas

The directionality of the emission for a single half-wave antenna is given by Equa-tion 5.18, and Figure 5.3 shows that it is only slightly more peaked along the θ = π /2direction than the proto-dipole oscillator. The emission can be made more directionalby implementing an array of half-wave dipole antennas along the x-axis as indicated in

λ/2 antenna

z

x

charge dipole

90˚78˚

Figure 5.3 Plot of the radiation distribution, normalised to themaximum at θ = π /2, from a linear half-wave antenna and singlepoint dipole. Angles 78◦ and 90◦ correspond to half-power pointsfor half-wave antenna and point dipole, respectively.

Real antennas 105

Z

P

θ

Y

X

d

φ

Figure 5.4 Five λ/2 antennas arranged in a linear array along thex-axis, each element of the array separated by a distance d.

Figure 5.4. The E-field of the array at any point is essentially the linear superpositionof the E-fields of the individual elements, each element offset by a phase shift related tothe array pitch d along x. Suppose we have N = 2n + 1 array elements symmetricallyarranged along ±x with n elements on each side of the origin. Then the field at point Pis simply,

Earray(P) = Ehw(P)m=n∑m=–n

e–imϕ

= Ehw(P)einϕm=(N–1)∑m=0

e–imϕ

= Ehw(P)einϕ1 – e–iNϕ

1 – e–iϕ

= Ehw(P)sin(Nϕ/2)sin(ϕ/2)

(5.20)

where ϕ is the azimuthal phase shift associated with the displacement a along x.Figure 5.5 shows the angular distribution of the E-field of a five-element linear arrayas shown in Figure 5.4:

ϕ = kxa =2πλ

sin θ cos ϕ a


z

x

Figure 5.5 Angular E-field distributionof a half-wave antenna array with fiveelements.

z

x10.4˚78˚ Figure 5.6 Angular distribution of radiated power

from the five-element array shown in Figure 5.4. Notethat the power directed along x with half-power pointsof 10.4◦ compared to a single half-wave antenna withhalf-power points at 78◦.

Figure 5.5 was determined from an oscillation frequency of ν = 1× 109 Hz and a wave-length λ = 0.3m. The separation between the elements of the array is d = λ/2 = 0.15m.We see that the maximum of the principal lobes of intensity occur for θ = π /2 but we seesmall adjacent lobes, reminiscent of a diffraction pattern. This result is hardly surprisingsince Equation 5.20 is the same as for linear diffraction grating of N elements.

The corresponding radiated power distribution of the five-element array is shown inFigure 5.6. The concentration of the radiated power along x is clearly evident. The an-gular aperture of the half-power points in the five-element array is only 10.4◦ comparedto 78◦ for the single-element antenna.

5.4 Summary

This relatively short chapter reintroduces point dipole radiation and expressions forthe energy flux and power radiated. The idea of radiative impedance and the imped-ance of free space is introduced. The discussion is then extended to real antennasin which the oscillating charge travels along lengths comparable to the wavelength.The important half-wave antenna, and arrays of half-wave antennas, then show howenhanced directionality can be achieved.

Further reading 107

5.5 Further reading

1. J. D. Kraus and R. J. Marhefka, Antennas for All Applications, 3rd edition, McGraw-Hill (2002).

2. J.-P. Pérez, R. Carles, and R. Fleckinger, Électromagnétisme Fondements et applications,Chapter 20, Masson (1997).

3. A. Sommerfeld, Partial Differential Equations in Physics, Chapter VI, Academic Press(1964).

6

Classical Black-body Radiation

6.1 Field modes in a cavity

Now that we have established the essential vocabulary of radiation fields and the equa-tions of motion governing them, we can begin our discussion of light–matter interactionby applying our new language to a straight forward problem from the classical theoryof radiation. What we seek to do is calculate the energy density inside a bounded con-ducting volume. We will then use this result to describe the interaction of light with acollection of two-level atoms inside the cavity.

The basic physical idea is to consider that the electrons inside the conducting volumeboundary oscillate as a result of thermal motion, and through dipole radiation, set upelectromagnetic standing waves inside the cavity. Because the cavity walls are conduct-ing, the electric field Emust be zero there. Our task is twofold: first to count the numberof standing waves that satisfy this boundary condition as a function of frequency; sec-ond, to assign an energy to each wave, and thereby determine the spectral distributionof energy density in the cavity.

After decoupling the electric and magnetic fields in Equations 2.27 and 2.28 we arriveat an expression that describes the equation of motion of an E-field wave,

∇2E =1c2∂2E∂t2

propagating in a charge-free space containing no E-field sources,

∇ · E = 0

Harmonic standing-wave solutions factor into oscillatory temporal and spatial terms.Now, respecting the boundary conditions for a three-dimensional box with sides oflength L, we have for the components of E:

Ex(x, t) = E0xe–iωt cos(kxx) sin(kyy) sin(kzz)

Ey(y, t) = E0ye–iωt sin(kxx) cos(kyy) sin(kzz) (6.1)

Ez(z, t) = E0ze–iωt sin(kxx) sin(kyy) cos(kzz)


Field modes in a cavity 109

where, again, k is the wave vector of the light, with amplitude

|k| = 2πλ

(6.2)

and components

kx =πnL

n = 0, 1, 2, ...

and similarly for ky, kz. Notice that the cosine and sine factors for the Ex field compo-nent show that the transverse field amplitudes Ey,Ez have nodes at 0 and L, as theyshould, and similarly for Ey and Ez. In order to calculate the mode density, we beginby constructing a three-dimensional orthogonal lattice of points in k space as shown inFigure 6.1. The separation between points along the kx, ky, kz axes is π

L , and the volumeassociated with each point is therefore

V =(πL

)3

Now the volume of a spherical shell of radius |k| and thickness dk in this space is 4πk2dk.However, the periodic boundary conditions on k restrict kx, ky, kz to positive values, sothe effective shell volume lies only in the positive octant of the sphere. The number ofpoints is therefore just this volume divided by the volume per point:

kz

ky

kx

kx

kz

ky

Lπ

θ

ϕ

Figure 6.1 Mode points in k space. Right panel shows one-half of thevolume surrounding each point. Left panel shows one-eighth of the volumeof spherical shell in this k space.

110 Classical Black-body Radiation

Number of k points in spherical shell =18

(4πk2dk

)(πL

)3 =12L3 k

2dkπ2

Remembering that there are two independent polarisation directions per k point, we findthat the number of radiation modes between k and dk is,

Number of modes in spherical shell = L3 k2dkπ2

(6.3)

and the spatial density of modes in the spherical shell is

Number of modes in shellL3

= dρ(k) =k2dkπ2

We can express the spectral mode density, mode density per unit k, as

dρ(k)dk

= ρk =k2

π2

and therefore the mode number as

ρkdk =k2

π2dk

with ρk as the mode density in k space. The expression for the mode density can beconverted to frequency space, using the relations

k =2πλ

=2πνc

=ω

c

and

dνdk

=c2π

Clearly,

ρνdν = ρkdk

and therefore

ρνdν =8πν2dνc3

The density of oscillator modes in the cavity increases as the square of the frequency.Now the average energy per mode of a collection of oscillators in thermal equilibrium,

Planck mode distribution 111

5.0500 K

500 K4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

5.0

0.00.0 0.2 0.4 0.6 0.8

ν (×1014) ν (×1014)

1.0 1.2 1.4 0.00.0

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1.0

1.0

1.2 1.4

300 K

300 K

200 K

200 K

ρRJ (ν)

(×10–17)E ρPl (ν)

(×10–18)E

Figure 6.2 Left panel: Rayleigh–Jeans black-body energy density distribution as a function offrequency, showing the rapid divergence as frequencies tend towards the ultraviolet (the ultravioletcatastrophe). Right panel: Planck black-body energy density distribution showing correcthigh-frequency behaviour.

according to the principal of equipartition of energy, is equal to kBT , where kB is theBoltzmann constant. We conclude, therefore, that the energy density in the cavity is

ρRJE (ν)dν =8πν2kBTdν

c3(6.4)

which is known as the Rayleigh–Jeans law of black-body radiation; and, as Figure 6.2shows, leads to the unphysical conclusion that energy storage in the cavity increasesas the square of the frequency without limit. This result is sometimes called the ‘ultra-violet catastrophe’ since the energy density increases without limit as oscillator frequencyincreases towards the ultraviolet region of the spectrum. We achieved this result bymultiplying the number of modes in the cavity by the average energy per mode. Sincethere is nothing wrong with our mode counting, the problem must be in the use of theequipartition principle to assign energy to the oscillators.

6.2 Planck mode distribution

We can get around this problem by first considering the mode excitation probabilitydistribution of a collection of oscillators in thermal equilibrium at temperature T . Thisprobability distribution Pi comes from statistical mechanics and can be written in terms

of the Boltzmann factor e–εi /kBT and the partition function q =∞∑i=0e–εi /kBT :

Pi =e–εi /kBT

q


Now, Planck suggested that instead of assigning the average energy kBT to every oscil-lator, this energy could be assigned in discrete amounts, proportional to the frequency,such that

εi =(ni +

12

)hν

where ni = 0, 1, 2, 3 . . . and the constant of proportionality h = 6.626 × 10–34 J·s. Wethen have

Pi =e–hν/2kBTe–nihν/kBT

e–hν/2kBT∞∑ni=0

e–nihν/kBT=

(e–hν/kBT

)ni∞∑ni=0

(e–hν/kBT

)ni (6.5)

=(e–hν/kBT

)ni (1 – e–hν/kBT)

(6.6)

where we have recognised that∞∑ni=0

(e–hν/kBT

)ni = 1/(1 – e–hν/kBT

). The average energy per

mode then becomes

ε =∞∑i=0

Piεi =

∞∑ni=0

(e–hν/kBT

)ni (1 – e–hν/kBT)(ni)hν =

hνehν/kBT – 1

(6.7)

and we obtain the Planck energy density in the cavity by substituting ε from Equation 6.7for kBT into Equation 6.4:

ρPlE (ν)dν =8πhc3ν3

1ehν/kBT – 1

dν (6.8)

This result, plotted in Figure 6.2, is much more satisfactory than the Rayleigh–Jeansresult since the energy density has a bounded upper limit and the distribution agreeswith experiment.

6.3 The Einstein A and B coefficients

Let us consider a two-level atom or collection of atoms inside the conducting cavity. Wehave N1 atoms in the lower level E1, and N2 atoms in the upper level E2. Light interactswith these atoms through resonant stimulated absorption and emission, E2 – E1 = hω0,the rates of which, B12ρω, B21ρω, are proportional to the spectral energy density ρωof the cavity modes. Atoms populated in the upper level can also emit light ‘spontan-eously’ at a rate A21 that depends only on the density of cavity modes (i.e. the volume

The Einstein A and B coefficients 113

of the cavity). This phenomenological description of light absorption and emission canbe described by rate equations first written down by Einstein. These rate equations weremeant to interpret measurements in which the spectral width of the radiation sourceswas broad compared to a typical atomic absorption line width, and the source spectralflux Iω (W/m2·Hz) was weak compared to the saturation intensity of a resonant atomictransition. Although modern laser sources are, according to these criteria, both narrowand intense, the spontaneous rate coefficient A21 and the stimulated absorption coeffi-cient B12 are still often used in the spectroscopic literature to characterise light–matterinteraction in atoms and molecules. These Einstein rate equations describe the energyflow between the atoms in the cavity and the field modes of the cavity, assuming ofcourse, that total energy is conserved:

dN1

dt= –

dN2

dt= –N1B12ρω +N2B21ρω +N2A21 (6.9)

At thermal equilibrium we have a steady-state condition dN1dt = – dN2

dt = 0 with ρω = ρthωso that

ρ thω =A21(

N1N2

)B12 – B21

and the Boltzmann distribution controlling the distribution of the number of atoms inthe lower and upper levels,

N1

N2=g1g2e–(E1–E2)/kT

where g1 and g2 are the degeneracies of the lower and upper states, respectively. So:

ρthω =A21(

g1g2ehω0/kT

)B12 – B21

=A21B21(

g1g2ehω0/kT

)B12B21

– 1(6.10)

But this result has to be consistent with the Planck distribution, Equation 6.8:

ρPlE (ν) dν =8πhc3

ν301

ehν0/kBT – 1dν (6.11)

ρPlE (ω) dω =h

π2c3ω30

1ehω0/kBT – 1

dω (6.12)

Therefore, comparing these last two expressions with Equation 6.10, we must have

g1g2

B12

B21= 1 (6.13)


and

A21

B21=

8πhc3ν30 (6.14)

or

A21

B21=hω3

0

π2c3(6.15)

These last two equations show that if we know one of the three rate coefficients, we canalways determine the other two.

It is worthwhile to compare the spontaneous emission rate A21 to the stimulatedemission rate B21:

A21

B21ρthω= ehω0/kT – 1 (6.16)

which shows that for hω0, much greater than kT (visible, UV, and X-ray), the spontan-eous emission rate dominates; but for regions of the spectrummuch less than kT (far IR,microwaves, and radio waves), the stimulated emission process is much more important.It is also worth mentioning that even when stimulated emission dominates, spontaneousemission is always present.

6.4 Summary

This chapter discusses black-body radiation by showing how correct mode counting in aconducting cavity leads to a disturbing conclusion if energy is partitioned equally amongall modes—the conclusion being that the energy density goes to infinity with increas-ing frequency. We then show how Planck saved the day (and energy conservation) bysuggesting the quantisation of energy packets; the size of which was proportional to thefrequency, and the population probability of which falls off exponentially. The chaptercloses with an identification of the phenomenological Einstein A and B coefficients withthe Planck distribution.

6.5 Further reading

1. T. S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894–1912, TheUniversity of Chicago Press (1978).

2. P. W. Milonni, The Quantum Vacuum, Academic Press (1994).

7

Surface Waves

7.1 Introduction

We treat here an important wave phenomenon: the surface wave. It occurs in manyphysical systems, both mechanical and electromagnetic. Mechanical surface waves canexist at the interface between two media with different densities, such as the oceans andearth. Ordinary sea waves and tsunamis are surface waves. Seismic events can produceboth longitudinal and shear waves at the earth’s surface, resulting in earthquakes. Elec-tromagnetic surface waves can occur at the interface between dielectric and conductivemedia or between two dielectrics satisfying appropriate boundary conditions. Air-saltwater or glass-metal interfaces support electromagnetic surface waves. Here we focus onelectromagnetic waves at the interface between common dielectrics and noble metals,such as air or glass and silver or gold. We will characterise these waves by their distinct-ive properties of dispersion, spatial field distribution, and polarisation. These propertieswill be used to analyse useful phenomena such as wave guiding within subwavelengthstructures and spatial light localisation below the diffraction limit.

7.2 History of electromagnetic surface waves

The impetus for the study of electromagnetic surface waves began with the developmentof radio communication near the end of the nineteenth century. Ordinary Maxwellian‘space’ waves cannot directly propagate beyond the horizon, althoughMarconi had beenable to demonstrate long-range radio communication, including transatlantic messagingbetween 1899 and 1901. A possible explanation for this successful over-the-horizontransmission, was the excitation and propagation of electromagnetic waves at the air-earth or air-sea interface that could be ‘guided’ over the curvature of the planet. Itwas therefore necessary to find solutions to Maxwell’s equations that could explainthe observations. Stimulated by Arnold Sommerfeld’s analysis of electromagnetic wavepropagation along a single conducting wire in 1899, his student Johann Zenneck pre-sented a paper on planar surface electromagnetic waves in 1907. Zenneck found thatMaxwell’s equations had a solution corresponding to a wave coupled to a flat surfaceinterface at the boundary between a dielectric (air) and a mediumwith finite conductivity


116 Surface Waves

VerticalDipole

~Medium 1Air

Medium 2Earth

z

y

x

R

Figure 7.1 Schematic arrangement for Zenneck wavegeneration.

(salt water). The configuration used by Zenneck is illustrated in Figure 7.1. Soon afterZenneck’s article appeared, Sommerfeld again published an influential paper in 1909in which he analysed the propagation of radio waves along the surface of the earth. Heconsidered a vertical oscillating dipole (radio antenna) perpendicularly positioned to theinterface between two media: one dielectric, the other conducting. Sommerfeld studiedsecondary waves produced by the discontinuity at the interface between air and land,considered a weakly conductive material. His analysis showed that the Hertz vector

that describes the electromagnetic field in each medium is given by the sum of two fields,P and Q, corresponding to two waves. The wave Q represents a space propagating waveand P a surface-guided wave. For long distances R from the source, Q varies as (1/R),where R =

√(r2 + z2) (see Figure 7.1 for a definition of the coordinates), while P var-

ies asymptotically as 1/√r and is confined to the earth’s surface. Over the course of

the next two decades a controversy arose over the existence of the so-called Zenneckwaves because Sommerfeld’s original analysis contained a sign error, and it was foundthat it was difficult to excite the surface-guided wave with practical antenna sources,or to detect and distinguish the true surface wave from the spatially propagating wavevery close to the surface. In any event, as far as long-distance wireless communicationwas concerned, the whole question became irrelevant when it was discovered that reflec-tion from the ionosphere was the principal agent responsible for over-the-horizon radiotransmission.

7.3 Plasmon surface waves at optical frequencies

Recent interest in electromagnetic surface waves stems from their potential applica-tions in nanoscale opto-electronic devices. In contrast to the Sommerfeld–Zenneckanalyses emphasising radio frequencies, modern studies at optical frequencies usuallytake Raether’s influential treatise, Surface Plasmons on Smooth and rough Surfaces and

Plasmon surface waves at optical frequencies 117

on Gratings, as their point of departure. The modern terminology for the Zenneckwave at the interface between a metal and a dielectric is the surface plasmon or thesurface plasmon-polariton (SPP). At optical frequencies the surface plasmon exhibitsstrong spatial localisation at the dielectric-conductor interface and can be simply, andefficiently, excited by appropriately polarised light incident on surfaces decorated withfeatures (slits, holes, and grooves, etc.) at the subwavelength scale (tens to hundreds ofnanometres). Here, we consider the interface between two media: one dielectric and theother a metallic conductor. Each material is characterised by a relative permittivity, εr:εdr for the dielectric, and εmr for the metal. In order to lighten notation we drop the sub-script r and write the relative permittivities (dielectric constants) as εd and εm for thedielectric and metal, respectively. The metal also exhibits a high conductivity, σ , andobeys ‘Ohm’s law’ E = σ J, where E is the electric field applied to the metal and J is thecurrent density induced in it. We consider the dielectric medium as essentially transpar-ent (lossless) so that εd = ε′d where ε′d is real. In the case of metals, absorptive loss inthe optical region of the electromagnetic spectrum is significant, the dielectric constantis complex, and εm = ε′m + iε′′m, where the imaginary term represents absorptive loss.Figure 7.2 shows two orthogonal linear polarisation orientations of E-M fields incidenton the metal surface.1 The plane of incidence is defined by the incident and reflectedplane-wave propagation vectors k, and in Figure 7.2, this plane is aligned with the x – zplane. WithH perpendicular to the plane of incidence (right-hand panel, Figure 7.2), thepolarisation is called ‘transverse-magnetic’ (TM). TM polarisation restricts the incidentwave to field components Ex,Ez,Hy. With E perpendicular to the same plane (left-handpanel, Figure 7.2), the polarisation is ‘transverse-electric’ (TE). The field components ofa TE polarised wave are Hx,Hz,Ey. We will consider continuity conditions at the surfacefor each case.

H

E

Direction of Propagation

k

y

z

TE

x

Ey

Hz

Hx

E

Ex

Direction of Propagation

k

y

z

TM

x

Ez

Hy

H

Figure 7.2 Configuration of field components for TE and TM polarised plane waves incident ona dielectric-metal interface.

1 Although in Chapter 2 the direction of propagation is always along z, in this chapter we must considerboth the incident propagating wave and the surface wave. The incident space-propagating wave is taken to beincident along z, while the surface wave propagates along x.

118 Surface Waves

We start with the two Maxwell curl equations, the Faraday law, and the Maxwell–Ampère law, Equations 2.27 and 2.28, and posit two solutions: E- and H-fieldscharacterised by the propagation vector k = (2π /λ)k, where λ is the wavelength, andoscillating harmonically in time with frequency ω:

E(r, t) = E0(r)ei(k·r–ωt) (7.1)

H(r, t) = H0ei(k·r–ωt) (7.2)

Application of the curl operation leads to the following relations:

∂Ey∂z

= –(iωμ0)Hx (7.3)

∂Hx

∂z–∂Hz

∂x= –i(ωε0ε)Ey (7.4)

∂Ey∂x

= (iωμ0)Hz (7.5)

TE polarisation

and

∂Hy

∂z= (iωε0ε)Ex (7.6)

∂Ex∂z

–∂Ez∂x

= (iωμ0)Hy (7.7)

∂Hy

∂x= –(iωε0ε)Ez (7.8)

TM polarisation

The incident and reflected waves propagate in the dielectric medium character-ised by the dielectric constant εd . The propagation vector in free space is denoted k0,and within the dielectric, kd = k0n, where n is the index of refraction. The dielectricconstant and the index of refraction of any medium are related by ε = n2. Since theincident k = ki is in the x – z plane we can write

k2i = k2x + k2z (7.9)

= εdk20 (7.10)

Similarly, on the metal side of the interface for the transmitted propagation vector km,we have

k2t = k2x + k2z (7.11)

= εmk20 (7.12)

7.3.1 Surface waves with TE polarisation

In TE polarisation we find the wave solution for Ey by applying the posited plane-waveexpression, Equation 7.1, to the relations Equations 7.3, 7.4 and 7.5. The result (in thedielectric medium) is


Edy (x, z) = Ed0yei[(kdxx+k

dzz)–ωt] (7.13)

From this E-field component, the H-field components on the dielectric side of theinterface are readily obtained:

Hdx = –

iωμ0

∂Edy∂z

=kdzωμ0

Edy (7.14)

Hdz =

iωμ0

∂Edy∂x

= –kdxωμ0

Edy (7.15)

The same analysis applies to the metal side of the interface with the result

Emy (x, z) = Em0yei[(kmx x+k

mz z)–ωt] (7.16)

and

Hmx = –

iωμ0

∂Emy∂z

=kmzωμ0

Emy (7.17)

Hmz =

iωμ0

∂Emy∂x

= –kmxωμ0

Emy (7.18)

At the interface, z = 0, continuity conditions for Hx and Ey require that

Hdx (z = 0) = Hm

x (z = 0) (7.19)

Edy (z = 0) = Emy (z = 0) (7.20)

and therefore, at the boundary,

kdz = kmz (7.21)√εdk0 =

√εmk0 (7.22)

But this last equation can only be valid for k0 = 0 since the dielectric constants on thetwo sides of the boundary can never be equal. Therefore there can be no wave solution atthe boundary for TE polarisation:

7.3.2 Surface waves with TM polarisation

With TM polarisation we have three field components, Hy,Ex,Ez. We find the wavesolution for Hy on the dielectric side of the interface:

Hdy (x, z) = Hd

y0ei[(kxx+kzz)–ωt] (7.23)

120 Surface Waves

Then using Equations 7.6 and 7.8, we find expressions for Edx and Edz :

Edx =kdz

ωε0εdHdy (x, z) (7.24)

Edz = –kdx

ωε0εdHdy (x, z) (7.25)

On the metal side of the interface we have

Emx =kmz

ωε0εmHmy (x, z) (7.26)

Emz = –kmx

ωε0εmHmy (x, z) (7.27)

Continuity conditions at the interface (z = 0) are

Edx (x, 0) = Emx (x, 0) (7.28)

Hdy (x, 0) = Hm

y (x, 0) (7.29)

εdEdz (x, 0) = εmEmz (x, 0) (7.30)

Then substituting Equations 7.24 and 7.25 and 7.26 and 7.27 into the continuityconditions we find that at the boundary

kdx = kmx = kx (7.31)

kdzεd

=kmzεm

(7.32)

The wave vector relations on the dielectric and metal sides of the interface, Equa-tions 7.10 and 7.12 allow us to write

(kmz )2

(kdz)2=εmk20 – (k

mx )

2

εdk20 – (kdx)2

(7.33)

Then, using Equations 7.31 and 7.32, we can eliminate the kz components and obtainan expression for ksx, the magnitude of the surface wave vector propagating along theboundary, in terms of the free-space wave vector k0 and the dielectric constants on eachside of the interface:

ksx =√

εdεm

εd + εmk0 (7.34)

The z components of the surface wave, on the dielectric and metallic sides of theboundary, can be easily obtained from


(ks,dz)2

= εdk20 –(ksx)2 k20 =

(1 –

εm

εm + εd

)εdk20 (7.35)

(ks,mz

)2 = εmk20 – (ksx)2 k20 =(1 –

εd

εm + εd

)εmk20 (7.36)

ks,dz = ± εd√εd + εm

k0 (7.37)

ks,mz = ± εm√εd + εm

k0 (7.38)

Note that if the real part of the dielectric constant in the metal is negative and ifεd + εm < 0, then kmz , k

dz will be imaginary. The Ez component of the surface wave on

both sides of the interface will then be evanescent:

ks,dz →±iκds = ±i∣∣∣∣ εd√εd + εm

∣∣∣∣ k0 (7.39)

ks,mz →±iκms = ±i∣∣∣∣ εm√εd + εm

∣∣∣∣ k0 (7.40)

The choice of sign (±) ensures that the wave amplitude decreases exponentially withincreasing distance (±z) from the interface. With the choice of coordinate axes as shownin Figure 7.2,

ks,dz → iκds z > 0 (7.41)

ks,mz → –iκms z < 0 (7.42)

and the components of the surface wave projecting onto the dielectric side take on thefollowing forms,

Hs,dy (x, z, t) = H0yeκ

ds zei(k

sxx–ωt) (7.43)

Es,dx (x, z, t) = –iκdsωε0εd

H0yeκds zei(k

sxx–ωt) (7.44)

Es,dz (x, z, t) = –ksx

ωε0εdH0yeκ

ds zei(k

sxx–ωt) (7.45)

Similar expressions obtain on the metallic side with the appropriate change of sign foriκms (Equation 9.104). Figure 7.3 shows how the electric and displacement fields of sur-face waves decrease exponentially on both dielectric and metal interfaces. In addition tothe exponential decrease in the ±z direction, the plots in Figure 7.3 also show that theabsolute magnitude of the dielectric constant on the metal side is typically much greaterthan on the dielectric side. Therefore, field penetration into the metal is limited to theskin depth. For gold or silver in the optical interval (350 ≤ λ0 ≤ 850 nm) of the elec-tromagnetic frequency spectrum, significant field penetration is of the order 20 – 30 nm.

122 Surface Waves

–700–0.5 0.5 1.5 2.5 3.5 4.5

Dielectric (air)

Dz(Z) (a.u.)

Dz(Z)

Ez(Z)

Ez(Z) (a.u.)

z(nm)

0

–600

–500

–400

–300

–200

–100

0

0 2

Metal (Ag)

4 6100

Figure 7.3 Vertical (z) components of the sur-face wave electric field (E-field) and displace-ment field (D-field). Note that Ez is discontinu-ous at the surface while Dz is continuous. Theunits of the E-field and D-field are arbitrary.

Furthermore, the E-field is discontinuous at the boundary, due to the presence of sur-face charge density, while the displacement field (D = ε0εE) takes into account thesign change in the dielectric constant across the boundary and is therefore continuous(although the derivative of the D-field with respect to the boundary normal is not con-tinuous). Table 7.1 summarises the various components of the electromagnetic wave inthe dielectric, the metal, and at the interface for TM polarisation.

Table 7.1 Field components for TM Polarisation

Region Ex(x, z, t) Ez(x, z, t) Hy(x, z, t)

dielectric kdzωε0εd

H(x, z, t) – kdxωε0εd

Hy(x, z, t) Hy0ei(kdxx+k

dzz–ωt)

metal kmzωε0εm

Hy(x, z, t) – kmxωε0εm

Hy(x, z, t) Hy0ei(kmx x+k

mz z–ωt)

interfaced side – iκds

ωε0εdHy(x, z, t) – ksx

ωε0εdHy(x, z, t) H0yeκ

ds zei(k

sxx–ωt)

m side iκmsωε0εm

Hy(x, z, t) – ksxωε0εm

Hy(x, z, t) H0ye–κms zei(k

sxx–ωt)

Equation 7.34 shows that ksx is complex since εm is complex. The three componentsof the surface wave Esx,E

sz,H

sy propagate in the x direction at the interface. Taking Esx as

an example,

Esx(x) = Es0xeiksxx = Es0xe

ik0nsx (7.46)


and the argument of the exponential will have a real and imaginary term since ksx = k0nsis complex. The factor ns = ηs + iκs is the complex surface index of the refraction atthe interface. The imaginary term represents the exponential loss in amplitude due todissipation in the metal as the wave propagates along the surface. From the definition ofthe surface index of refraction, ns = ksx/k0, the complex dielectric constant of the metal,εm = ε′m + iε′′m and Equation 7.34, we can determine the corresponding complex index ofrefraction:

ns =√

εmεd

εm + εd(7.47)

n2s =ε′mεdε′m + εd

·1 + i ε

′′mε′m

1 + i ε′′mε′m+εd

(7.48)

For the two metals commonly used in plasmonic devices, silver and gold,∣∣ε′′m/ε′m∣∣ 1

over the optical range of interest, and this characteristic can be used to simplify Equa-tion 7.48. Figure 7.4 and Figure 7.5 show a plot of the real and imaginary terms of thedielectric constant of silver metal as a function of wavelength over the optical range,and Figure 7.6 plots the ratio of

∣∣ε′′m/ε′m∣∣ 1 for the same silver data. Separating Equa-tion 7.48 in real and imaginary terms while dropping fractional terms quadratic in ε′′m,we find that

n2s =ε′mεdε′m + εd

·[1 + i

ε′′mεdε′m(ε′m + εd

)]

(7.49)

5

–15

–35

–55

–75

–95

–115

–135

–155

–175

–195

–215200 400 600 800 1000

λ (nm)

ε´(λ)

1200 1400 1600 1800 2000

Figure 7.4 Real part of the dielectric constant ε′ vs. wavelength forsilver from the Johnson and Christy measurements (see Reference 8 inthe reading list in Section 7.10).

124 Surface Waves

8

7

6

5

4

3

2

1

0200 400 600 800 1000

λ (nm)

ε˝(λ)

1200 1400 1600 1800 2000

Figure 7.5 Imaginary part of the dielectric constant ε′′ vs.wavelength for silver from the Johnson and Christy measurements(Reference 8. in Section 7.10).

–0.100 400 800 1200 1600 2000

–0.09

–0.08

–0.07

–0.06

–0.05

–0.04

–0.03

–0.02

–0.01

0.00

λ (nm)

ε˝ε´

Figure 7.6 Ratio ε′′/ε′ vs. wavelength for silver from the Johnsonand Christy measurements (Reference 8 in Section 7.10).


and from the definition of the complex index of refraction,

n2s = η2 – κ2s + i2ηκs (7.50)

Setting the real and imaginary terms in Equations 7.49 and 7.50 equal,

η2s – κ2s =

ε′mεdε′m + εd

(7.51)

2ηsκs =ε′mε′′mεd

ε′m(ε′m + εd

)2 (7.52)

Equations 7.51 and 7.52 can be decoupled and solved for ηs and κs, the two componentsof the surface complex index of refraction. The result is:

ηs =

√ε′mεdε′m + εd

(7.53)

κs =12ε′′m(ε′m)2(ε′mεdε′m + εd

)3/2

(7.54)

These expressions can be simplified further if the dielectric is vacuum or air. In thatcase, εd can be set equal to unity and

ηs =

√ε′m

ε′m + 1(7.55)

κs =1

2√ε′m· ε′′m(ε′m + 1

)3/2 (7.56)

As a typical example of a surface wave at a metal-dielectric interface, we takeλ0 = 632 nm, εd = 1 (air), εm = –15.625 + i(1.04059) (silver), and calculate ηs, κs. FromEquations 7.55 and 7.56 and the definition of k0 we find that

ηs = 1.0336

κs = 2.3534× 10–3

k0 = 2π /λ0 = 9.942× 106 m–1

Taking the ‘effective length’ xeff to be the distance over which the amplitude of the surfacewaves falls to 1/e of the initial value, we find xeff = 42.74μm. Figure 7.7 shows thebehaviour of the surface wave Ex-field amplitude at the interface (z = 0), propagatingin the x direction. Figure 7.8 shows the variation of Ex, Ez near the metal-dielectricinterface.

126 Surface Waves

0 0.4 0.8 1.2 1.6 2.0 2.4 2.8

Ez(x,0)

|Ez(x,0)|

3.2 3.6 4.0

x (nm) (×104)

Figure 7.7 The surface wave Ex-field amplitude at the interface (z = 0),propagating in the x direction. The distance along which the amplitudedecreases to 1/e of the initial value at x = 0 is called the ‘effective’ length.

z

x

Esw(x,z)

λsw

– +++ ––

Figure 7.8 Instantaneous distribution of the electric field Ex and Ez inthe x and z directions, respectively. The signal variation of the field Ezalong x is accompanied by the variation of electric charge (+,-)corresponding to the condition of Ez-field discontinuity. The representationof surface charge density along z is exaggerated for clarity. The chargedensity can only exist at the surface.

7.4 Plasmon surface wave dispersion

7.4.1 The free-electron plasma

The dielectric constant of materials in general depends on the frequency of the elec-tromagnetic fields with which they interact. The equation describing this dependence is

Plasmon surface wave dispersion 127

called the dispersion relation, and the free-electron plasma model leads to a very simpledispersion relation for high-conductivity metals. The force on a free electron subject toan applied E-field in one dimension is

med2xdt2

= –eEx (7.57)

and if the E-field is time-harmonic, E = E0e–iωt, we have:

– ω2mex = –eEx (7.58)

The polarisation field P of the free-electron plasma, considered as a volume density ofone-dimensional dipoles px = –ex, is

P = –Nee2

meω2E (7.59)

withNe the electron concentration of the plasma. From the definition of the displacementfield D, and the constitutive relation between D and E, we have

D = ε0E + P (7.60)

D = ε0εE (7.61)

and therefore the dielectric constant of the free-electron plasma is

ε(ω) = 1 +Pε0E

= 1 –Nee2

ε0meω2(7.62)

The characteristic plasma frequency is defined by

ω2p =

Nee2

ε0me(7.63)

So finally, we have the lossless dielectric constant:

ε(ω) = 1 –ω2p

ω2(7.64)

7.4.2 The Drude model of metals

The free-electron plasma model describes harmonic electron motion in the limit of anegligible restoring force acting on the electron, and no ‘collisional’ or dissipative losses.The Drude model does not introduce a finite restoring force, but does add dissipation tothe equation of motion for the electrons in the plasma gas (when a linear restoring force

128 Surface Waves

is introduced into the conduction electron equation of motion, the theory is called theDrude–Lorentz model of metals). The equation of motion for the simple Drude modelof the conduction electrons can still be described by a one-dimensional driven harmonicoscillator expression, where the driver issues from an electromagnetic field incident atthe dielectric-metal boundary. If we assume the incident radiation is polarised along x,then the Drude model equation of motion can be written as

med2xdt2

+me�dxdt

= –eE(x, t) = –eE0me–iωt (7.65)

where me is the electron mass, � is a phenomenological damping constant, and eE0me–iωt

is the driving force of an E-M field in the metal with amplitude E0m and frequencyω. Again, dropping the common oscillatory phase term, the solutions for the position,velocity, and acceleration are:

x =1

me(ω2 + i�ω

) eE0m (7.66)

dxdt

= –iω

me(ω2 + i�ω

) eE0m (7.67)

d2xdt2

= –ω2

me(ω2 + i�ω

) eE0m (7.68)

Again, we write the polarisation as a density of dipoles, Px = –Neex, and now thedielectric constant is complex:

ε(ω) = 1 –Nee2

ε0me(ω2 + i�ω

) = 1 –ω2p

ω2 + i�ω(7.69)

Separating the real and imaginary terms,

ε(ω) =ω2 – ω2

p + �2

ω2 + �2+ i

ω2p�

ω(ω2 + �2)(7.70)

If � ω, as is the usual case at optical frequencies,

ε(ω) = ε′(ω) + iε′′(ω)

ε(ω) � 1 –ω2p

ω2+ iω2p�

ω3(7.71)

A summary of the Drude-model parameters of many common metals is given inTable 7.2.


Table 7.2 Drude-model parameters for some common metals∗

Metal ωp (rad/s) � (s–1) τ (s) Ne (m–3)

Al 2.24× 1016 2.73× 1013 3.66× 10–14 1.56× 1029

Ag 1.37× 1016 5.56× 1013 1.80× 10–14 5.83× 1028

Au 1.37× 1016 4.05× 1013 2.47× 10–14 5.83× 1028

Co 6.03× 1015 5.56× 1013 1.80× 10–14 1.13× 1028

Cu 1.12× 1016 1.38× 1013 7.25× 10–14 3.90× 1028

Fe 6.22× 1015 2.77× 1013 3.61× 10–14 1.20× 1028

Mo 1.13× 1016 7.77× 1013 1.29× 10–14 3.97× 1028

Ni 7.43× 1015 6.63× 1013 1.51× 10–14 1.72× 1028

Pb 1.12× 1016 3.07× 1013 3.25× 10–14 3.90× 1028

Pd 8.29× 1015 2.34× 1013 4.28× 10–14 2.14× 1028

Pt 7.82× 1015 1.05× 1013 9.51× 10–14 1.90× 1028

Ti 3.83× 1015 7.20× 1013 1.39× 10–14 4.56× 1027

V 7.84× 1015 9.22× 1013 1.08× 10–14 1.91× 1028

W 9.75× 1015 9.18× 1013 1.09× 10–14 2.96× 1028

∗ Compiled from data in Reference 8. in Section 7.10

7.4.2.1 Drude model dispersion curves

The dispersive response to electromagnetic excitation is characterised by the frequencydependence of the dielectric constant of a material, ε(ω). Many common optical dielec-trics (air, glass, and fused quartz) exhibit dielectric constants with near-zero imaginaryterms (negligible absorption) and real terms that vary little with frequency over the vis-ible range. We have just seen how metallic response can be described by simple physicalmodels such as the free-plasma model or the Drude model. The behaviour of a materialis usually summarised by the ‘dispersion curve’ that plots the energy or frequency of theexciting radiation against the propagation constant k of the light in the material. Thereis a simple chain of relations between the propagation constant k(ω) and the dielectricconstant ε(ω). First,

k =2πλ

and λ =λ0

n(7.72)

where λ0 is the free-space wavelength and n the index of refraction. Then, the materialindex of refraction can be complex, n = η + iκ, and is related to the dielectric constantε by

n(ω) =√ε(ω) (7.73)

η(ω) + iκ(ω) =√ε′(ω) + iε′′(ω) (7.74)

and therefore,

η2 – κ2 = ε′(ω) (7.75)

2ηκ = ε′′(ω) (7.76)

130 Surface Waves

Decoupling η and κ results in a quadratic equation in η:

η4 – η2ε′(ω) –14

[ε′′(ω)

]2= 0 (7.77)

The solution to Equation 7.77 can be obtained directly from the elementary quad-ratic formula. The real part of the index of diffraction η can then be substituted intoEquation 7.76 to obtain the imaginary part κ. Thus η, κ, and ε′, ε′′ can be readily inter-converted. Figure 7.9 shows the dielectric constant terms and the refractive index terms,calculated from Eq. 7.71 as a function of ω/ωp, where ωp is the plasma frequency forsilver metal, ωp = 1.35×1016 s–1, and the relaxation rate � is taken to be � = 1×1014 s–1.

The dispersion curve for a Drude metal can be obtained by starting with the ex-pressions for the real and imaginary parts of the dielectric constant, Equation 7.71, andrescaling the frequency variable to y = ω/ωp:

ε′m = 1 –1y2

(7.78)

ε′′m =1y3

(�

ωp

)(7.79)

and taking the propagation constant of the surface wave along the dielectric-metal inter-face as ks = k0ns. The factor ns is the surface of index of refraction, the real andimaginary parts of which we write from Equations 7.53 and 7.54:

ωp

Γ= 7.41×10–3

ωp

Γ= 7.41×10–3

2.00(a)

ε´ε˝

К

(b)

1.50

1.00

0.50

0.000.50 1.00

ω/ωp ω/ωp

1.50

n

2.000 0.50 1.00 1.50 2.000

1.90.0

–3.8

–7.6

–11.4

–15.2

–19.0

–22.8

1.6

1.2

0.8

0.4

0.0

Figure 7.9 Real (panel a) and imaginary (panel b) terms of the index of refraction and the dielectricconstant calculated from the Drude model (Equation 7.71) vs. frequency ω normalised to ωp, theplasma frequency for silver. The relaxation rate � is taken to be � = 1014 s–1 andωp = 1.35× 1016 s–1.


ks = k0ns = k0

[(εdε′m

εd + ε′m

)1/2

+ iε′′m

2(ε′m)2

(εdε′′m

εd + ε′m

)3/2]

(7.80)

ks = k′s + ik′′s (7.81)

Thus:

k′s = k0

(εdε′m

εd + ε′m

)1/2

(7.82)

k′′s =ε′′m

2(ε′m)2

(εdε′′m

εd + ε′m

)3/2

(7.83)

At first neglecting the imaginary term, we define a new ‘reduced’ variable:

x =k′scωp

=k′skp

(7.84)

and write Equation 7.82 as:

x = y(

εd(y2 – 1)y2(εd + 1) – 1

)1/2

(7.85)

Now Equation 7.85 results in a quadratic equation in y, the solution of which is

y =

√√√√εd + (εd + 1)x2 ±√(εd + (εd + 1)x2

)2– 4εdx2

2εd(7.86)

The solution constitutes the dispersion curve (in ‘reduced’ units) of the frequency vs. thepropagation parameter for the surface wave. The reduced variable x is defined in termsof the real part of the propagation parameter k′s, and therefore, this expression describesonly the real part of the dispersion curve. The solution consists of two branches, plottedin Figure 7.10. The lower branch describes the surface-wave dispersion from the longwavelength limit (y → 0; x → 0) to an asymptote (y → ysp = ωsp/ωp; x → ∞). Thisfrequency asymptote at the short wavelength limit corresponds to the surface plasmaresonance where the surface charge density oscillates collectively over the entire surfaceand the phase velocity of the surface wave dω/dks slows to near zero. An expression forthis surface plasma asymptotic frequency can be obtained from Equation 7.86:

ω→ ωsp =ωp√1 + εd

(7.87)

This condition is often called the surface plasmon resonance and bears a simple relationto ωp, the bulk plasmon resonance derived from the free-electron plasma model. Between

132 Surface Waves

00.0

0.5

1.0

1.5

Light-line

1 2k s/kp

3 4 5

ω/ωp

ωp

ωs

Figure 7.10 Dispersion curve for a silver-glass interface. Thereal dielectric constant for glass is taken to be ε′ = 2.25. Theloss term in the metal is not included.

these two frequencies there is no charge density surface wave or bulk wave propaga-tion. This ‘forbidden zone’ is sometimes termed a stopband. At frequencies above ωp,propagation once again occurs on the surface and in the bulk. This frequency limit cor-responds to the point where ε′m → 0. Below ωp, ε′m < 0 and above ωp, ε′m > 0. Also shownin Figure 7.10 is the linear ‘light line’, the dispersion curve of light freely propagating inthe dielectric medium, ωll = k0c/η.

Inclusion of the imaginary part of the surface parameter ks in the definition of thereduced variable x in Equation 7.84, and subsequent solution of Equations 7.85 and7.86, modifies the dispersion curve and opens propagation in the stopband. The realand imaginary parts of the full dispersion curve are shown in Figure 7.11. Note that inpanel (a) of this figure, the phase velocity

vphase =ω

k′s(7.88)

reverses sign in the stopband region. This property is a necessary characteristic of‘negative index’ metamaterials. Panel (b) shows that, unfortunately for many potentialapplications, this region is also one of high absorption.

7.4.2.2 Conduction current in a Drude metal

The conduction current in the metal is given by

Jc = eNedxdt

= eNeω

me(�ω – iω2

) eE0m (7.89)


00.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

GlassGlass

(b)(a)

1 2 3 4 5 0 1 2 3 4 5

εd = 2.25εd = 2.25

k s/kpk s/kp

ω/ωp ω/ωp

ωp

ωsp

ωp

ωsp

Figure 7.11 Panel (a): dispersion curve for the real part of the surface propagationparameter k′s. Panel (b): Dispersion curve for the imaginary part of the surfacepropagation parameter k′′s . Both propagation parameters have been normalised tokp = ωp/c. Inclusion of the imaginary term results in finite propagation in thestopband (compare with Figure 7.10).

and with the electron density related to the bulk plasma frequency, Equation 7.63, wehave

Jc =ω2p

(� – iω)ε0E0m (7.90)

From the standard constitutive relations

Jc = σE =σ

ε0ε0E (7.91)

and so, comparing Equations 7.90 and 7.91, the expression for the complex conductivitycan be expressed as

σ

ε0=

�(�2

ω2p+ ω2

ω2p

) + iω(

�2

ω2p+ ω2

ω2p

) (7.92)

We can compare the damping rate � to the plasma frequency ωp if we can estimatethe latter, and from Equation 7.63 we see that this requires knowledge of the conduc-tion electron density. A typical ‘good’ metal, such as silver, exhibits an electron densityNe � 6× 1028 m–3, and therefore, ωp � 1.5× 1016 s–1. The damping due to radiative orcollisional loss is typically ∼ 1014 s–1, and therefore, � ωp. Thus, in the optical regimewe can write

σ

ε0� �ω2

p

ω2+ iω2p

ω(7.93)

134 Surface Waves

This last expression for the conductivity shows that at relatively low frequencies (RF,microwave, and radio wave) the real term dominates, and the conductivity is in phasewith the driving field and ‘ohmic’. As the frequency increases into the optical range,however, the electrons can no longer follow the driving field in phase, and the imaginaryterm dominates. At ω � 1015 s–1, typical of the visible optical regime, the real term isonly about 5 per cent of the imaginary term. Figure 7.10 plots the dispersion relation ofthe Drude model, Equation 7.71.

7.5 Energy flux and density at the boundary

7.5.1 Energy flux normal to the boundary

As indicated in Equation 2.61, the Poynting vector is defined as

S = E×H (7.94)

and the energy flux of an electromagnetic wave averaged over an optical cycle is givenby

S =12Re[E×H∗] (7.95)

Figure 7.12 shows the x and z components of the energy flow on the two sides of thesurface between a dielectric and a metal. Writing out these components explicitly fromTable 7.1, we have, at normal incidence, the transmitted energy flux impinging at thesurface from the dielectric into the metal along the z direction:

Sdz =12Re[EdxH

∗y

]=

12Re[

kdzεdε0ω

]H2

0y (7.96)

Smz =12Re[Emx H

∗y

]=

12Re[

kmzεmε0ω

]H2

0y (7.97)

DielectricSz

d

Szm

Sxd

Sxm

metal

z

x

Hy(x,z)

Figure 7.12 Indication of the components of the Poynting vector of thedielectric-metal interface in which a wave propagates surface plasmonpolariton (SPP).

Energy flux and density at the boundary 135

From the continuity relation at the surface, Equation 7.33, we see immediately thatthe energy flux is continuous across the boundary in the z direction. On the dielectricside, the permittivity ε0εd and the propagation constant kdz are real (for a lossless mater-ial), and therefore, the expression for the energy flux is also real. On the metal side thedielectric constant is complex; εm = ε′m + iε′′m, as is kmz = k0nm = k0(ηm + iκm). The metalindex of refraction nm is related to the metal dielectric constant through nm =

√εm, and

therefore, the expression for the product of Emx and H∗y in Equation 7.97 exhibits realand imaginary terms. The real part of Equation 7.97 represents the transmitted, expo-nentially decreasing energy flux into the metal. In order to obtain explicit expressionsfor nm and kzm we need to examine the propagation of electromagnetic waves inside realmetals.

7.5.2 Electromagnetics in metals

We start by writing the curl equations of Maxwell in terms of E and B, the electric fieldand magnetic induction field:

∇ × E = –∂B∂t

Faraday law (7.98)

∇ ×B = μmεm∂E∂t

+ μmJc Maxwell–Ampère law (7.99)

where μm and εm are the permeability and permittivity of the metal, and Jc is the freecharge current. For a high-conductivity metal such as silver or gold, Jc is directly propor-tional to E with the conductivity σ , and the proportionality constant, Jc = σE. Equations7.98 and 7.99 can be decoupled by first applying the curl operation to the Faradayequation,

∇ × (∇ × E) = ∇(∇ · E) – ∇2E = ∇ ×(∂B∂t

)= –

∂(∇ ×B)∂t

(7.100)

= –μmεm∂2E∂t2

– μmσ∂E∂t

(7.101)

which together with ∇ · E = 0, results in a wave equation for E in the metal:

∇2E = μmεm∂2E∂t2

+ μmσ∂E∂t

(7.102)

Similarly, applying the curl operation to Equation 7.99 produces the wave equationfor B:

∇2B = μmεm∂2B∂t2

+ μmσ∂B∂t

(7.103)

136 Surface Waves

Plane-wave solutions to these equations for waves propagating in the z direction can bewritten as

Em = E0mei(kmz z–ωt) (7.104)

Bm = B0mei(kmz z–ωt) (7.105)

The amplitudes E0m,B0m and the propagation vector kmz are generally complex quan-tities. Substituting these solutions back into the uncoupled wave equations, Equa-tions 7.102 and 7.103, results in an expression for the propagation parameter in termsof the permeability, complex permittivity, and complex conductivity of the metal,(

kmz)2 = μmεmω2 + iμmσω (7.106)

with

kmz = kr + iki εm = εr + iεi σ = σr + iσi (7.107)

Equating real and imaginary terms after substitution into Equation 7.106 yields

k2r – k2i = μ0εrω

2 – μ0σiω (7.108)

2krki = μ0εiω2 + μ0σrω (7.109)

where we have set μm = μ0. Now writing the complex permittivity in terms of thedielectric constants, εr = ε0ε1 and εi = ε0ε2, we have

k2r – k2i = μ0ε0ω

2(ε1 –

σi

ε0ω

)= k20

(ε1 –

σi

ε0ω

)(7.110)

2krki = μ0ε0ω2(ε2 +

σr

ε0ω

)= k20

(ε2 +

σr

ε0ω

)(7.111)

where μ0ε0 = 1/c2 and k0 = ω/c have been used. Setting

k20

(ε1 –

σi

ε0ω

)= β2 (7.112)

k20

(ε2 +

σr

ε0ω

)= γ 2 (7.113)

we separate Equations 7.110 and 7.111:

kr = ± β√2

√√√√1±

√1 +

(γ

β

)4

(7.114)

ki = ± β√2

√√√√–1±

√1 +

(γ

β

)4

(7.115)

Energy flux and density at the boundary 137

But the complex propagation parameter km can also be expressed in terms of thecomplex index of refraction, nm:

km = k0nm = k0(ηm + iκm) (7.116)

and nm is related to the metal dielectric constant εm through

nm =√εm =

√ε′m + iε′′m (7.117)

Again, equating real and imaginary terms,

ηm =

√ε′m2

√√√√1±√1 +

(ε′′mε′m

)4

(7.118)

κm =

√ε′m2

√√√√–1±√1 +

(ε′′mε′m

)4

(7.119)

and consequently

kr = k0ηm = k0

√ε′m2

√√√√1±√1 +

(ε′′mε′m

)4

(7.120)

ki = k0κm = k0

√ε′m2

√√√√–1±√1 +

(ε′′mε′m

)4

(7.121)

Comparing Equations 7.120 and 7.121 with Equations 7.114 and 7.115 shows that

ε′m =(ε1 –

σi

ε0ω

)(7.122)

ε′′m =(ε2 +

σr

ε0ω

)(7.123)

Equations 7.122 and 7.123 separate the metal dielectric constant into two parts: first ε1and ε2 represent the dispersive and absorptive response of the metal excluding the con-duction electrons, and second σi,r/ε0ω represents the conducting electrons contributionto the dielectric constant. Substituting for σi,r /ε0ω from Equation 7.93, we have

ε′m =

(ε1 –

ω2p

ε0ω2

)ε′′m =

(ε2 +

�ω2p

ε0ω3

)(7.124)

138 Surface Waves

If we consider the metal response, excluding the conduction electrons, to behave essen-tially as a lossless dielectric material, we can posit ε1 and ε2 as 1 and 0, respectively.This assumption is equivalent to the Drude model in the high frequency limit,Equation 7.71.

7.5.3 Energy flux along the boundary

The energy flux of the surface wave propagating along the x direction:

Ss,dx =12Re[Es,d ×Hs,d] = 1

2Es,dz

(Hdy

)∗=

12

ksxεdε0ω

H20y (7.125)

Ss,mx =12Re [Es,m ×Hs,m] =

12Es,mz

(Hmy

)∗=

12Re[

ksxεmε0ω

H20y

](7.126)

Note that the real part of the metal permittivity is usually a relatively large negativenumber (see Figure 7.4, for example), and therefore, as shown in Figure 7.12, Ss,mzhas the opposite sign from Sx,dz . The net energy flux is the sum of the Ss,dx and Ss,mxcomponents.

7.6 Plasmon surface waves and waveguides

7.6.1 Introduction

The plasmon surface wave at a metal-dielectric interface can be considered a guidedwave since it follows the surface. In this sense the interface itself acts as a one-dimensional (1-D) waveguide. We will take up a systematic discussion of transmissionlines and waveguides in Chapter 8. Here we show that plasmon-like electromagneticwaves can propagate in the presence of two parallel interfaces separated by a sub-wavelength gap and obtain the properties of this ‘gap’ plasmon propagating mode.Figure 7.13 shows a slab waveguide with two exterior metallic cladding layers and oneinterior dielectric layer. The lowest order propagating mode of this guide resembles asurface-wave, SPP, bounded by the two outer metal claddings instead of the usual sin-gle surface. Sometimes this propagating mode is called a ‘gap plasmon’. Other modes,symmetric (cosine-like) or antisymmetric (sine-like) with respect to the core centreline,are possible as the width of the core increases. If the two metal cladding layers aredifferent (e.g. gold and silver), the guided wave will not be perfectly symmetric or anti-symmetric since the dispersion curves of different metals vary. The allowed modes canbe determined by matching the fields in the transverse z direction at the cladding-coreboundaries. The matching condition results in a transcendental equation that can besolved numerically for the propagation parameter q as a function of the core width a.Here we carry out in some detail the matching operation for TMmodes as an illustrativeexample.

Plasmon surface waves and waveguides 139

Transverse fielddistribution

Propagationa

z

x

Be–rz

C cos(qx+ϕ)

ε3

ε2

ε1

Aepz

Figure 7.13 Design of a planar waveguide formed by three zones ofdifferent permittivity: ε1, ε3, the ‘cladding’ or shell can be consideredmetallic, and ε2, the ‘core’, a dielectric. This type of structure is sometimestermed a MIM waveguide (metal-insulator-metal). The complementarystructure, IMI waveguides, are also possible. Propagating modes can besymmetric or antisymmetric with respect to the core centreline.

7.6.2 Field matching at core-cladding boundaries

7.6.2.1 TM polarisation

For TM (transverse magnetic) polarisation we have three field components, Hy,Ex,Ez,in each of the three zones. From the coordinate setup in Figure 7.13 we see that Hy andEx are parallel to the boundaries, and therefore, must be continuous across them. Weassume a harmonic force-field wave F propagating in the x direction:

F(r, t) = F(y, z)ei(kxx–ωt) (7.127)

and use the two curl equations of Maxwell to find relations among the relevant fieldcomponents. We further assume that the materials are non-magnetic with relativepermeability μ = 1:

∂Hy

∂z= iωε0εEx (7.128)

∂Ex∂z

– ikEz = iωμ0Hy (7.129)

ikHy = –iωε0εEz (7.130)

Now we write down the real field components in the transverse z direction in each of thethree zones indicated in Figure 7.13, while expressing the propagation in the x directionas eikx:

140 Surface Waves

For z < 0

Hy = Beikxepz (7.131)

Ex =–iBωε3ε0

peikxepz =(

–ipωε3ε0

)Hy (7.132)

Ez =–Bkωε3ε0

eikxepz =(

–kωε3ε0

)Hy (7.133)

For 0 < z < a

Hy = Aeikx cos(qz + ϕ) (7.134)

Ex =i

ωε2ε0qAeikx sin(qz + ϕ) (7.135)

Ez =–k

ωε2ε0Aikx cos(qz + ϕ) =

(–k

ωε2ε0

)Hy (7.136)

For z > a

Hy = Cikxe–rz (7.137)

Ex =iCωε1ε0

reikxe–rz =(

irωε1ε0

)Hy (7.138)

Ez =–Ckωε1ε0

eikxe–rz =(

–kωε1ε0

)Hy (7.139)

Next we write the continuity conditions at each boundary for Hy and Ex.

For z = 0

B =A cosϕ (7.140)(–ipωε3ε0

)B =

(iq

ωε2ε0

)A sinϕ (7.141)

For z = a

A cos(qa + ϕ) =Ce–ra (7.142)iq

ωε2ε0A sin(qa + ϕ) =C

irωε1ε0

e–ra (7.143)

Rearranging and dividing Equation 7.141 by Equation 7.140 we have:

tanϕ = –ε2pε3q

(7.144)

and proceeding similarly with Equations 7.143 and 7.142 we have:

tan (qa + ϕ) =tan qa + tanϕ1 – tan qa tanϕ

=ε2rε1q

(7.145)

Plasmon surface waves and waveguides 141

Substituting the right-hand side of Equation 7.144 into Equation 7.145 results in thetranscendental equation, the solutions to which define the propagating modes in thewaveguide:

tan qa =ε2rε1q

+ ε2pε3q

1 –ε22pr

ε1ε3q2

(7.146)

The propagation parameters, p, q, and r are also related through

∂Ex∂z

– ikEz = iωμ0Hy (7.147)

Substituting the relevant field components into this expression shows that

k2 – p2 = ε3k20 (7.148)

q2 + k2 = ε2k20 (7.149)

–r2 + k2 = ε1k20 (7.150)

where k0 = ω/c is the propagation parameter of the free-space propagating wave with fre-quency ω. Rearranging these expressions, we can define an ‘effective’ index of refractionthat characterises the propagating mode in the core:

p =√k2 – ε3k20 = k0

√n2eff – n

23 (7.151)

q =√ε2k0 – k2 = k0

√n22 – n

2eff (7.152)

r =√k2 – ε1k20 = k0

√n2eff – n

21 (7.153)

where n1 =√ε1, n2 =

√ε2, n3 =

√ε3, and neff = k/k0.

7.6.2.2 TE polarisation

For TE (transverse electric) polarisation the relevant field components are Ey,Hx,Hz,and carrying through the matching procedure at z = 0, z = a, we find the transcendentalequation,

tan qa =rq +

pq

1 – prq2

(7.154)

As qa → 0 we find a limiting expression in TM polarisation, Equation 7.146, for thechannel width a,

a = –ε1p + ε3rε2pr

(7.155)

142 Surface Waves

which is physically allowed since ε1, ε3 are negative. Therefore, even as a → 0, apropagating mode always exists. In the case of TE polarisation the limiting expression is

a = –p + rpr

(7.156)

and since the channel width cannot be negative, this result indicates that TE modes arenot supported in the deep subwavelength limit.

7.7 Surface waves at a dielectric interface

7.7.1 Introduction

In the previous sections of this chapter we have focused almost entirely on electromag-netic surface waves at dielectric-metal interfaces, and we have developed the physicsof charge density waves as the source of these surface plasmon-polariton fields. Wehave also observed that the interface between two media can also be considered a 1-Dwaveguide. In the present section we will again use boundary matching, as we did inSection 7.6.2, to find wave solutions confined to the line defined at the interface be-tween two dielectric slabs, one of which is also in contact with a ground plane. Wewill also find the 1-D waveguide modes of these surface waves for both TM and TEpolarisation.

The geometry is shown in Figure 7.14. We will consider the top dielectric, unboundedin the positive x direction, as air or vacuum; and the dielectric slab with thickness t mightbe glass or silicon. This arrangement can be realised practically on ridge waveguides andis similar to stripline or microstrip geometries popular in integrated circuit design. Westudy TM and TE modes separately and assume surface wave propagation in the +zdirection with propagation parameter β.

Ground plane

z

x

t

ε0

εd

Figure 7.14 Surface waves propagate along z at the interfacebetween vacuum ε0 and material dielectric εd. The ground planetruncates wave propagation in –x direction and sets field amplitudesto zero at the εd-ground plane interface.

Surface waves at a dielectric interface 143

7.7.2 TM modes

TM polarisation specifies the existence of Ex,Ez,Hy polarised amplitudes. We seek solu-tions to the scalar wave equation in the region above the ε0/εd interface and in the regionwithin the εd dielectric slab. We match these solutions at the interface.

Above the interface the free-space propagation parameter k0 in the x–z plane is relatedto kx and β by k20 = k2x + β

2, and the 1-D wave equation in the space above the interfaceis given by

(∂2

∂x2+ k20 – β

2)

Ez(x) = 0 (7.157)

and within the dielectric slab,

(∂2

∂x2+ k2d – β

2)

Ez(x) = 0 (7.158)

where k2d = εdk20 and Ez = Ez(x)eiβz. We assume that there is no variation in the fieldsalong the y direction. Within the dielectric slab, the general solution, in terms of linearcombinations of complex travelling waves, is

Ez(x) = Aeikcx ± Be–ikcx (7.159)

where

k2c = k2d – β2 = εdk20 – β

2 (7.160)

Above the interface the E-field of the surface wave must decrease exponentially, andthe form of the solution must be

E az (x) = Ce–κx (7.161)

where

κ2 = β2 – k20 (7.162)

and E az (x) is the phasor amplitude in the air above the dielectric slab. We anticipate that

the propagation velocity of the surface wave will be somewhat slower than a wave ofthe same frequency propagating in air since a significant fraction of the surface waveamplitude will be immersed in the dielectric slab. Therefore, we expect β > k0 andEquation 7.162 is written so that κ is real and can be chosen with a positive sign. Thischoice assures that Equation 7.161 represents an exponentially decaying amplitude inthe +x direction.

144 Surface Waves

At x = 0 the boundary condition requires E dz (0) = 0. Therefore, B = A and

E dz (x) = ±A

(eikcx – e–ikcx

)= ±A2i sin kcx (7.163)

where E dz (x) is the amplitude for the surface-wave phasor Ez(x) = Ez(x)eiβz. At x = t the

amplitudes for the E-field amplitudes on both sides of the interface have to match:

± A2i sin kct = Ce–κt (7.164)

The H-field amplitudes have to match as well, and we obtain the H-fields from the E-fields by using the relations obtained in Chapter 8, Section 8.5.1. In the dielectric slabwe have

H dy (x) =

ik2cωε0εd

∂Ez∂x

= ± ikcωε0εdA2i cos kcx (7.165)

and on the air side

H ay (x) = –

iκωε0Ce–κx (7.166)

At the interface, H-field amplitudes must match. Since H ay (x) has a negative amplitude,

we must choose the negative amplitude solution for H dy (x):

–ikcωε0εdA2i cos kct = –

iκωε0Ce–κt (7.167)

Now divide Equation 7.164 by Equation 7.167, the two matching equations at theinterface. The result is

kc tan kct = εdκ (7.168)

which provides a relation between the wave vector kc, the thickness of the dielectric slabt, and the damping constant κ, for a given slab material with dielectric constant εd .Equations 7.159 and 7.160 specify that kc is the wave vector component along x of thatpart of the surface wave within the dielectric slab. We can write κ in terms of kc and k0by using the fact that the propagation parameter β along z must be the same for the twoparts of the surface wave: one part in the dielectric slab and the other in the air. We cantherefore eliminate β from Equations 7.160 and 7.162 resulting in the expression

k2c + κ2 = k20(εd – 1) (7.169)


In principle we could eliminate κ between Equations 7.168 and 7.169, and obtain anexpression for kc as a function of three parameters: the free-space wave vector k0, thematerial dielectric constant εd , and the slab thickness t. However, Equation 7.168 is atranscendental equation with no simple analytic solution for kc. The best we can do isfind solutions for kc numerically and plot them to gain some insight into the propertiesof the TM modes.

One way to plot the solutions is to divide both sides of Equation 7.168 by kc andrearrange it as

tan(kct) =εdκ

kc(7.170)

and consider kc as an independent variable y. We can then plot both sides of the equation,and identify solutions to the transcendental equation as the points of intersection:

y1 = tan(kct) (7.171)

y2 =εdκt(kct)

=εd[k20(εd – 1) – k

2c

]1/2t

(kct)(7.172)

Figure 7.15 shows, schematically, the appearance of the two families of curves y1 and y2when plotted against kct. Clearly, y1 is just a plot of the tangent function with branchesstarting at y1 = 0 for kct = 0,π , 2π . . .. It is also clear that only positive values of y1can provide legitimate solutions to Equation 7.168 since the numerator of y2 consists ofthree factors, all of which are intrinsically positive. For fixed εd and k0, y2, as a functionof (kct), is a family of curves parametrically increasing with the slab thickness t. FromEquation 7.172 we see that y2 → ∞ as kct → 0 and that y2 crosses 0, becoming pureimaginary, when kc = k0

√εd – 1. Intersections of y2 with y1 within the positive half of

the first branch (0 ≤ kct ≤ π /2) constitute solutions to the transcendental equation,Equation 7.168, belonging to the lowest TM mode, TM0. We see immediately that theTM0 mode has no cutoff for kc as kct → 0. For fixed kc, t and as k0 increases, y2 willbegin to intersect y1 both in the first and in the second positive branch (π ≤ kct ≤ 3π /2).These intersections in the second branch constitute TM1 solutions to Equation 7.168.Higher order TM modes begin to propagate as the kc = k0

√εd – 1 upper limit increases

with k0. From Figure 7.15 it is clear that the cutoff points for TM1, TM2, . . . occur atkct = nπ with n = 1, 2, 3, . . . . The general expression for the cutoff frequency (in Hzunits) is given by,

νc =kc

2π√με

(7.173)

where 1/√με is the local, ‘effective’ speed of light. For the TM modes of this surface

wave, therefore, we have:

νc =nc

2t√εd – 1

n = 0, 1, 2, . . . (7.174)

146 Surface Waves

10

8

6

4

2

0

–20.0 0.5 1.0 1.5 2.0 2.5 3.0

t = 100 nm t = 1000 nm

TM modes

tan(kct)

y1 = tan(kct)

y2(kct)

kct (units of π/2)

Figure 7.15 The expressions y1, y2 plotted as a function of kct. Thefunction y2 is plotted for a range of thicknesses t from 100 nm to1000 nm. Dashed curves show four representative solutions where thethickness t permits modes TM0 and TM1. Note that TM0 has nocutoff thickness but TM1 cuts off at kct = π . Intersection of the curvesy1, y2 > 0 constitute solutions to the transcendental equation,Equation 7.170.

where εd – 1 represents an ‘effective’ dielectric constant for the TM modes in the 1-Dsurface wave.

7.7.3 TM field solutions

Once kc has been found for a fixed set of parameters k0, t, εd , the TM fieldsEz,Ex,Hy can be obtained from Equations 7.163 and 7.164, and the relations inSection 8.5.1:

Ez(x, z) = ±A′ sin(kcx)eiβz 0 ≤ x ≤ t (7.175)

Ez(x, z) = ±A′ sin(kct)e–κ(x–t)eiβz x ≥ t (7.176)

where the constant A′ replaces the constant factors in Equation 7.163, A′ = A2i.


For Ex we find that

Ex(x, z) = ± iβkc A′ cos(kcx)eiβz 0 ≤ x ≤ t (7.177)

Ex(x, z) = ∓ iβκA′ sin(kct)e–κ(x–t)eiβz x ≥ t (7.178)

and for Hy that

Hy(x, z) = ± iωε0εdkcA′ cos(kcx)eiβz 0 ≤ x ≤ t (7.179)

Hy(x, z) = ∓ iωε0κ

A′ sin(kct)e–κ(x–t)eiβz x ≥ t (7.180)

7.7.4 TE modes

In contrast to the case of plasmon surface waves, the 1-D waveguide between two dielec-trics can support TE waves as well as TM waves. The fields specified by TE polarisationare Hz,Hx,Ey, and the scalar wave equation above the dielectric slab is written as

(∂2x∂x2

– κ2)

Hz = 0 (7.181)

and, since the dielectric slab is non-magnetic, within the slab the wave equation is

(∂2x∂x2

+ k20 – β2)

Hz = 0 (7.182)

The form of the solutions within the slab is

H dz (x) = Aeikcx ± B–ikcx (7.183)

with k2c = k20 – β2. On the air side above the slab we again posit an exponentiallydecreasing amplitude:

H az (x) = Ce–κx (7.184)

Again, at x = 0, the E-field parallel to the ground plane must vanish. Obtaining theE-field from the H-field using the relations in Section 8.5.1, we have

E dy (x) = –

ik2cωμ0

∂Hz

∂x

= ∓ωμ0

kcA2i sin kcx (7.185)

148 Surface Waves

and in the air space above the dielectric slab,

E ay (x) =

iκωμ0Ce–κx (7.186)

The expression for E dy in Equation 7.185 implies that the general solution for H d

z (x) inEquation 7.183 specialises to

H dz (x) = ±A2 cos kcx (7.187)

and that, therefore, the H-field is maximum at x = 0, the ground-plane boundary. At firstthought this condition might seem perplexing since no field should exist in the groundplane. What the boundary condition really means is that an infinitesimal distance abovethe boundary, the H-field amplitude is ±2A, and an infinitesimal distance below theboundary, the H-field is null.

At the boundary x = t, Hz and Ey must match:

±A2 cos kct = Ce–κt (7.188)

∓ωμ0

kcA2i sin kct =

iκωμ0Ce–κt (7.189)

Dividing Equation 7.188 by Equation 7.189 and rearranging results in

– cot kct =κ

kc(7.190)

Again, we set

y1 = – cot kct (7.191)

y2 =κtkct

=

√k20(εd – 1)

k2c– 1 (7.192)

and plot y1, y2 with kct the independent variable. Solutions to Equation 7.190 are atthe positive intersections of y1, y2. The general form of the solutions is shown in Fig-ure 7.16. In the first zone 0 ≤ kct ≤ π /2, no TE modes are possible because all the y1solutions are negative and therefore unphysical. The first TE mode, TE1, appears inthe zone π /2 ≤ kct ≤ π . From Figures 7.15 and 7.16 we see that the energy orderingof the first three modes is TM0, TE1, and TM1. Subsequent TE modes first appear atkct = 3π /2, 5π /2, 7π /2 . . .. The cutoff frequencies are then:

νc =nc

4t√εd – 1

n = 1, 3, 5, . . . (7.193)


10

8

6

4

2

0

–20.0 0.5 1.0 1.5 2.0 2.5 3.0

t = 100 nm

t = 1000 nm

TE modes

kct (units of π/2)

y1 = –cot(kct)

–cot(kct)

y2(kct)

Figure 7.16 The expressions y1, y2 plotted as a function of kct. Thefunction y2 is plotted for a range of thicknesses t from 100 nm to1000 nm. The first four dashed curves indicate representative values ofy2(kct) that do not admit solutions. Cutoff for the first allowed TEmode (TE1) occurs at kct = π /2. Intersection of the curves in the ydomain where y1, y2 > 0 constitute solutions to the transcendentalequation, Equation 7.190.

7.7.5 TE field solutions

Once kc has been found for a fixed set of parameters k0, t, εd , the TE fields Hz,Hx,Eycan be obtained from Equations 7.184 – 7.187 and the relations in Section 8.5.1:

Hz(x, z) = ±2A cos(kcx)eiβz 0 ≤ x ≤ t (7.194)

Hz(x, z) = ±2A cos(kct)e–κ(x–t)eiβz x ≥ t (7.195)

For Hx(x, z):

Hx(x, z) = ∓ βkc i2A sin(kcx)eiβz 0 ≤ x ≤ t (7.196)

Hx(x, z) = ± iβκ2A cos(kct)e–κ(x–t)eiβz x ≥ t (7.197)

150 Surface Waves

and for Ey(x, z):

Ey(x, z) = ∓ iωμ0

kc2A sin(kcx)eiβz 0 ≤ x ≤ t (7.198)

Ey(x, z) = ± iωμ0

κ2A cos(kct)e–κ(x–t)eiβz x ≥ t (7.199)

As we can see from Figures 7.15 and 7.16, the energy ordering of the first three modesis TM0, TE1, and TM1. Figures 7.17 and 7.18 show two illustrative examples of thedielectric guided waves for a structure in which a 200 nm silicon slab is sandwiched

–5–5

–4

–4

–3

–3

–2

–2

–1

–1.5

–0.5

0.5

1

1.5

0

–1

–1

0

1

1

2

2

3

3

4

4

5

5

×10–6

×10–7 ×10–3

0X-Axis (m)

Hy-field Si Waveguide, PEC Ground Plane, TE Polarization, λ0 = 1550 nm

λX = 692 nmTE, PEC Ground Plane PEC

Si200 nm

Air

Y-A

xis

(m)

Figure 7.17 Guided waves propagating along x in a 200 nm silicon slab. The figure shows the Hycomponent in a TE mode. The top layer is air and the bottom layer is a ground plane perfect electricconductor (PEC). The guided waves are launched from a 100 nm wide slit at the centre of thestructure. The source is a total-field-scattered-field (TFSF) source with λ0 = 1550 nm impinging onthe slit along the y (vertical) axis. The effective wavelength of the guided wave is λx = 692 nm, andtherefore the effective refractive index of the guiding structure is neff = 2.24. The two black verticallines on the left-hand side are reference markers for measuring the effective wavelength. Note that thez- and x-axes are reversed from Figure 7.14.

Summary 151

–5 –4 –3 –2

x

y

–1 1 2 3 4 5×10–6

0

X-axis (m)

λ = 650 – 657 nmTE, Ag Ground Plane

–5

–4

–3

–2

–1

0

1

2

3

4

5

Y-a

xis

(m)

×10–7 ×10–3Hy, Si Waveguide, Ag Ground Plane, TE Polarization,

–1.5

–0.5

0.5

1

1.5

0

–1Ag

Si200 nm

Air

Figure 7.18 Guided waves propagating along x in a 200 nm silicon slab. The figure shows the Hycomponent in a TE mode. The top layer is air and the bottom layer is a ground plane of silver (Ag).The guided waves are launched from a 100 nm wide slit at the centre of the structure. The source isa total-field-scattered-field (TFSF) source with λ0 = 1550 nm impinging on the slit along the y(vertical) axis. The effective wavelength of the guided wave is λx = 654 nm, and therefore theeffective refractive index of the guiding structure is neff = 2.37. The black vertical lines on theleft-hand side are reference markers for measuring the effective wavelength. Note that the z- andx-axes are reversed from Figure 7.14.

between air and a ground plane or ‘perfect electric conductor’ (PEC). The figures areplots of numerical solutions to Maxwell’s equations using the finite-difference, time-domain (FDTD) method.

7.8 Summary

The chapter begins with a brief historical sketch of the importance of surface waves tothe early days of radio communication before launching into the development of TEpolarised surfaces (that do not exist) and TM polarised surface waves (that do). Thedispersion models of the free-electron gas and the Drude metal are then introduced.The component and net Poynting vectors are then presented both for lossless and lossy

152 Surface Waves

(real) metals. Surface waves are guided waves, and in anticipation of the chapter onwaveguides, the first discussion of boundary wave matching is worked out. The chapterends with a discussion of waves at the boundary of dielectrics subject to a ground plane.This situation is reminiscent of the ‘stripline’ geometry used in the electrical engineer-ing of micro-electronics. The waves at the top surface of the stripline are evanescent,and in this sense can be considered ‘surface waves’ although they are really part of thepropagating waves inside the Si waveguide.

7.9 Exercises

1. It is a well-known fact that aluminium (Al), soon after exposure to air, exhibits atransparent aluminium oxide layer (Al2O3). This oxide layer is about 100 nm thickand has an index of refraction nAl2O3 = 1.77. What would be the effective in-dex of refraction of a surface plasmon polariton (SPP) wave propagating at themetal/metal oxide interface if the incident light has a wavelength of 400 nm? Theindex of refraction of Al at = 400 nm is n = 0.381 + i4.883.

2. A SPP wave propagates at the interface between air and a gold (Au) layer. Calculatethe phase and group velocity if the incident wavelength is 780 nm. In the com-mon case of good conductors where

∣∣ε′m∣∣ � ε′′, the group velocity is, to goodapproximation,

vg � Re

⎡⎣c{√

εdε′mεd + ε′m

+ω

2·[εdε′m

εdε′m

]3/2·[

1

ε′2m

(dε′mdω

)+

1

ε′d2

(dε′ddω

)]}–1⎤⎦

(7.200)and from the Drude model the complex dielectric constant for Au can be expressedas

ε′m(ω) = 1 –(ωpω

)2(7.201)

ε′′m =

(ω2p

ω3τ

)(7.202)

The bulk plasmon resonance frequency is denoted by ωp, and τ is the characteristicdamping time in the metal. For Au the values are ωp = 1.37 × 1016 s–1 and τ =2.47× 10–14 s.

3. Calculate the spatial dependence of the electric and magnetic field amplitudes ofan SPP wave propagating at the interface between a silver (Ag) layer and a glasssubstrate. The incident wave is 780 nm. Use the Drude model of metals to calculatethe permittivity of Ag at the given wavelength. The bulk plasma frequency of Ag isωp = 1.40 × 1016 rad/s and the relaxation rate is � = 1.0 × 1014 s–1. The dielectricconstant of glass is taken to be ε = 2.25.

Further reading 153

4. With reference to the preceding problem, calculate the SPP energy flux along thepropagation direction (x) at two points: x = 0 and x = 100nm.

5. An SPP wave propagates along the x direction at an air-gold interface. The free-space wavelength of the incident wave exciting the SPP is 632 nm, and the origin ofthe SPP wave is at x = 0. Calculate the spatial extent of the SPP amplitude alongthe z direction, normal to the interface, at x = 0. Determine the 1/e points for theamplitude and intensity on the air side and on the metal side of the interface.

6. Suppose a plane wave (free-space wavelength λ = λ0) is normally incident on adielectric-metal interface. Consider the energy density at the interface between thetwo media: a dielectric with permittivity ε0εd and a metal with permittivity ε0εm.Write the relation between the electric and magnetic energy densities on the dielectricside and the metal side of the interface. Assume the dielectric is lossless but εm =ε′m + iε′′m.

7. In Section 7.7 we developed the properties of surface waves propagating at the inter-face between two dielectrics, one of which was grounded. Follow the same procedureof field matching at the boundaries to find the surface waves at an air-dielectric inter-face where the dielectric is in contact with a perfect electrical conductor (PEC). Findthe transcendental equations for TM polarisation, analogous to Equation 7.168.


1. A. Sommerfeld, Partial Differential Equation in Physics, Lectures on TheoreticalPhysics Vol. VI, Chap. VI, Problems of Radio Academic Press, New York (1964).

2. A. Sommerfeld, Ueber die fortpflanzung elektrodynamischer wellen längs eines drahte.Ann Phys Chem vol 67, pp. 233–290 (1899).

3. J. Zenneck, Fortplfanzung ebener elektromagnetischer Wellen längs einer ebenen Leiter-fläche. Ann Phys vol 23, pp. 846–866 (1907).

4. H. Raether, Surface plasmons on Smooth and Rough Surfaces and on Gratings. Springer,Berlin (1988).

5. M. Born and E. Wolf, Principles of Optics, 6th edition, Pergamon Press (1993).

6. T-I. Jeon and D. Grischkowsky, THz Zenneck surface wave (THz surface plasmon)propagation on a metal sheet. Applied Physics Letters, vol. 88, p. 061113 (2006).


8. P. B. Johnson and R. W. Christy, Optical Constants of the Noble Metals, Phys Rev Bvol 6, pp. 4370–4379 (1972).

9. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, Op-tical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au,Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V and W. Appl Optics vol 24, pp. 4493–4499(1985).

8

Transmission Lines and Waveguides

8.1 Introduction

In Chapter 2 we summarised the elements of electromagnetics in terms of electric andmagnetic vector force fields related by Maxwell’s equations. We saw in Section 2.4 thatthe two time-varying equations, Faraday’s law, Equation 2.27 and the Maxwell–Ampèrelaw, Equation 2.28, give rise to electromagnetic wave solutions that propagate througha medium or, as we saw in Chapter 7, at the interface between media. These solutionsare generally distributed throughout a spatial extent that is large with respect to thecharacteristic wavelength.

In contrast, the conventional radio and microwave circuit theory of electrical en-gineering considers time-varying electromagnetic phenomena from the standpoint ofvarious combinations of lumped elements (capacitors, inductors, and resistors), localisedin space, and linked by interconnections of negligible impedance. Furthermore, the spa-tial extent of these circuit elements, together with their sources of voltage and current,oscillating from tens of kilohertz to hundreds of megahertz, is usually much smaller thanthe characteristic wavelength. The result is that all the elements of the circuits are subjectto the same time dependence without retardation. This time behaviour is termed quasi-static, and the physical arrangement of the circuit can be said to be subwavelength. Thesubwavelength scaling similarity between dielectric and metallic nanostructures inter-acting with optical driving fields and electrical circuits driven by conventional voltageand current oscillators suggests that circuit analysis might find useful application in thedesign of functional plasmonic and photonic devices. The main goal of this chapter,therefore, is to explore this possibility.

8.2 Elements of conventional circuit theory

Circuit theory is essentially the application of Maxwell’s equations to problems com-monly encountered in electrical engineering. The conditions that validate circuit theoryare: (1) that electrical effects happens instantaneously throughout the circuit—the quasi-static approximation; (2) that the net charge on every component is null; and (3) that


Elements of conventional circuit theory 155

+

+

+

+

–

–

–

–

R

C

I

L

c

e

d

h

b

g

fa

Vg

Figure 8.1 Schematic of an electrical circuitshowing lumped elements of resistor, inductor, andcapacitor in series with a voltage source. Also shownis the direction of current relative to the polarity ofthe voltage source.

magnetic coupling between or among lumped components of the circuit are negligible.A typical circuit comprising a source, resistor, capacitor, and inductor are shown inFigure 8.1.

8.2.1 Kirchhoff ’s rules

The starting point in circuit analysis are the two rules annunciated by Gustav Kirchhoffin 1845. The rules are as follows. (1) The scalar sum of the electric potential differencesVi around any closed circuit loop is null:∑

i

Vi = 0 (8.1)

(2) The scalar sum of charge currents Ii flowing out of a junction node is null:∑i

Ii = 0 (8.2)

8.2.1.1 Kirchhoff’s voltage rule

We consider first the voltage rule, which is the more relevant of the two for our purposes.Equation 8.1 derives from the differential form of Faraday’s law. By applying Stokes’theorem to Equation 2.27 we have∫

S(∇ × E) · dS =

∮E · dl = –

∫S

∂B∂t· dS (8.3)

where E, B are the usual electric and magnetic induction fields, S is the outward surfacenormal through which the curl of E protrudes, and l is the line around the boundary ofS, the positive direction of which is taken by using the right-hand rule. With respect toFigure 8.1, we take the line integral around the circuit and define the voltage differencebetween two reference points a, b as

Vba = –∫ b

aE · dl (8.4)

156 Transmission Lines and Waveguides

Then using Equation 8.3 we have

–∫ b

aE · dl –

∫ d

cE · dl –

∫ f

eE · –

∫ h

gE · dl =

∫S

∂B∂t· dS =

∂

∂t

∫SB · dS (8.5)

or

V0(t) + Vdc + Vfe + Vhg =∂

∂t

∫SB · dS (8.6)

where V0(t) is the voltage source connected to terminals a, b, Vdc is the resistive volt-age drop, and Vfe and Vhg the inductive and capacitive voltage drops, respectively. Inorder to identify Equation 8.6 with the Kirchhoff rule we have to make two assump-tions: (1) that there is no changing magnetic flux traversing the plane of the circuitloop. In simple circumstances where we only consider R,L,C lumped circuits, this con-dition is not hard to realise. Even if there is a changing magnetic field present, it canusually be incorporated into the inductive voltage term to make an ‘effective’ voltagedrop. (2) Voltage drops along the wire connecting the lumped elements in the loop arenegligible.

8.2.1.2 Resistance

We know from Ohm’s law the current density J through a conductor is proportional tothe E-field along the conductor, J = σE, where σ is the conductivity. Then the voltagedrop across the resistor is

Vdc = –∫ d

cE · dl = –

∫ d

c

Jσ· dl (8.7)

We also know that∫J · dl can be written in terms of the total current I flowing through

the conductor and the resistivity ρ, the inverse of the conductivity:

Vdc = –∫ d

c

IAρ dl = –IR (8.8)

where the resistance R is defined as

R =∫ d

c

ρ

Adl (8.9)

and A is the cross-sectional area of the conductor. The familiar expression Equation 8.8is valid at low frequencies where the current flows through the conductor uniformly overA. At high frequencies the current in not uniform across A, passing only near the surfaceand within the skin depth.

Elements of conventional circuit theory 157

8.2.1.3 Induction

Here we apply Faraday’s law to the local inductive element labelled L in Figure 8.1 (notthe whole circuit loop). The integral has to be over a closed path linking the terminalse, f . The first branch of the loop goes through the current-carrying coil from e to f . Thesecond branch is the return path from f to e along an arbitrary path in the space outsidethe coil. This second branch contributes nothing to the voltage drop across the coil:

∮E · d l =

∫ f

eE · d l +

[∫ e

fE · d l = 0

](8.10)

Then,

Vfe =∫ f

eE · dl = –

∂

∂t

∫SB · dS (8.11)

With the definition of the inductance L as the integral of the magnetic induction B overthe surface surrounding the inductive element, per unit current I passing through thecoil of the inductor, we have

L =

∫B · dSI

(8.12)

Therefore:

Vfe = –∂(LI)∂t

= –L∂I∂t

(8.13)

8.2.1.4 Capacitance

We assume that the capacitive element can be represented by a parallel plate capacitor.From elementary electrostatics we know that the definition of capacitance is the totalcharge Q accumulated on one plate divided by the voltage Vhg across the plate:

C =QV

(8.14)

But

Q =∫I dt (8.15)

and thus the voltage drop across the capacitor in terms of the capacitance C and thecurrent looping through the circuit is

Vhg = –Vgh = –

∫I dtC

(8.16)


Putting all three lumped circuit elements together with the positive voltage sourceVba = Vs and using the Kirchhoff voltage rule, we have

Vs – IR – LdIdt

–1C

∫I dt = 0 (8.17)

8.2.1.5 Kirchhoff’s current rule

Kirchhoff ’s current rule, that the charge currents Ii flowing into and out of a circuitjunction sum to zero, is illustrated in Figure 8.2. The basis for Equation 8.2 is essentiallythe charge conservation relation that derives from the Maxwell–Ampère equation

∇ ×H = J +∂D∂t

(8.18)

Taking the divergence of both sides of this equation, and remembering that div · curl = 0we find that

∇ · J + ∂∇ ·D∂t

= 0

∇ ·(J +

∂D∂t

)= 0 (8.19)

i3

i1

i2

i4

+–

Figure 8.2 Circuit junction point from which currents enter andleave. The Kirchhoff current rule states that the sum of thecurrents into and out of the junction must be zero.

Transmission lines 159

Then using Stokes’ theorem we can write∫A

∇ ·(J +

∂D∂t

)dA =

∮c

(J +

∂D∂t

)· dl = 0 (8.20)

Equation 8.20 says that the sum of the conduction current density J and displacementcurrent density ∂D/∂t around a closed loop c defining a surface A is null. From a circuitpoint of view, the closed loop can be considered any loop around a junction point intoand out of which currents flow along highly spatially localised wires. The integral overthe conduction current can be considered a sum over the current Ii of wire i by writing∮

cJ · δ(li – l) dl =

∑i

Ii (8.21)

For harmonically time-varying circuits the second term on the right in Equation 8.20can be written as ∮

c

∂D∂t· dl =

∮c–iωD · dl � 0 (8.22)

since at radio and microwave frequencies the displacement current is negligible com-pared to the conduction current. At optical frequencies the displacement current mightbecome significant and in this regime Kirchhoff ’s current rule has to be checked on acase-by-case basis. We shall assume as a working assumption that Kirchhoff ’s currentrule is valid and write ∮

c

(J +

∂D∂t

)· dl �

∑i

Ii � 0 (8.23)

8.3 Transmission lines

8.3.1 Lumped-element circuit analysis of a transmission line

In conventional circuit analysis, lumped circuit elements such as resistors, inductors,and capacitors are assumed to be dimensionally ‘subwavelength’, i.e. much smaller thanthe characteristic wavelength of the driving field. In contrast, a transmission line is acircuit entity transporting voltage and current waves over distances much greater thana wavelength. However, Kirchhoff ’s two rules can be used to analyse a transmissionline as a succession of lumped circuit elements. A two-wire transmission line extendingalong the z direction can be represented by the diagram in Figure 8.3 that shows onesegment of a repeating circuit laid out along �z. The lumped elements R,L, and Crepresent the resistance, inductance, and capacitance per unit length, respectively. Thefourth quantity, G, represents the ‘shunt conductance’ per unit length due to dielectricabsorption between the two conductors. The series resistance in the two conductors isdue to the finite (but very high) conductivity of the metal, while shunt conductance


RΔz LΔz

I(z)

I(z)

GΔz CΔzV(z + Δz)

V(z)

++

−−

Δz

Figure 8.3 Lumped circuit element representation of atransmission line repeating segment with length �z. Thetransmission line extends along the z direction.

is due to the finite (but very low) conductivity of the insulating dielectric. These fourquantities divide into two groups: R and G are dissipative and represent loss, while Land C are reactive and represent stored energy.

We can use the Kirchhoff rules to analyse this circuit segment. Proceeding around thecircuit and applying the voltage rule (Equation 8.1) we have

V (z, t) – R�zI(z, t) – L�z∂I(z, t)∂t

– V (z +�z) = 0 (8.24)

while the current rule (Equation 8.2) gives

I(z, t) –G�zV (z +�z, t) –C�z∂V (z +�z)

∂t– I(z +�z, t) = 0 (8.25)

Now divide both equations by �z and take the limit as �z→ 0. The result is

∂V∂z

= –RI(z, t) – L∂I(z, t)∂t

(8.26)

∂I∂z

= –GV (z, t) –C∂V (z, t)∂t

(8.27)

As usual we assume harmonic time variation so the partial derivative terms with respectto t simplify to

d Vd z

= – [R – iωL] I(z) (8.28)

d Id z

= – [G – iωC]V (z) (8.29)


These two equations can be uncoupled to yield

d2V (z)d z2

– β2V (z) = 0 (8.30)

d2 I(z)d z2

– β2I(z) = 0 (8.31)

with

β2 = (R – iωL)(G – iωC) (8.32)

β = ±√(R – iωL)(G – iωC) (8.33)

and

β ≡ iβ – γ (8.34)

The solutions are very reminiscent of the plane wave solutions of Chapter 3, Section 3.1.In the positive z space the voltage and current solutions look like damped travellingwaves propagating in the forward z direction. The voltage wave forward travelling takesthe form

V = V +0 e

βz = V+0 e

i(β+iγ )z (8.35)

where β is the propagation parameter1 and γ the dissipation term. In the negative zspace we have backwards travelling waves:

V = V –0 e

–βz = V –0 e

–i(β+iγ )z (8.36)

The general solution is some linear combination of forward and backward travellingwaves:

V = V+0 e

βz + V –0 e

–βz (8.37)

Usually the amplitude of the backward propagating wave V –0 will not be equal to the

amplitude of the forward wave V+0 , and if the backward wave is reflected (the usual

case), the forward and backward amplitudes will be related by a reflection coefficient.From Equation 8.28 we can write I(z) in terms of V+

0 and V –0 :

I(z) = I+0 + I –0 = –dV /dzR – iωL

= –iβ – γR – iωL

[V+

0 – V –0

](8.38)

1 Common usage in optics and electromagnetics is to denote k for the propagation parameter. Intransmission line theory, developed by and for electrical engineering, the term β is more common.


In analogy to Equation 3.28 we define an impedance Z0 as the ratio of the forwardvoltage amplitude to the forward current amplitude:

Z0 =V +

0

I+0(8.39)

Substituting from Equation 8.38, and using Equations 8.32 and 8.34, we have

Z0 = –R – iωLiβ – γ

=

√R – iωLG – iωC

(8.40)

where we have used the negative root of β2 from Equation 8.33 in the denominator ofEquation 8.40. The transmission line current can now be written in terms of the voltageand the characteristic impedance:

I(z) =1Z0

[V+

0 eβz + V –

0 e–βz]

(8.41)

In most practical transmission lines R and G are very small, and in the limit of a losslesstransmission line we find

Z0→√LC

(8.42)

8.3.2 Lossless plane-parallel transmission line

If we start with a lossless transmission line as the point of departure, we can calculatethe inductance per unit length, a function of the transmission line geometry, directlyfrom Faraday’s law. Figure 8.4 shows the magnetic flux lines between two parallel platescarrying a constant current I in opposite directions. The separation between the platesis given by s and the width of the plates is w. The ratio of w/s is sufficiently great that themagnetic flux density B is mostly concentrated between the plates and can be consideredconstant. The magnetic field between the plates can be related to the current flowingin one of the plate conductors through the integral form of the Maxwell–Ampère law,Equation 2.28. Using Stokes’ theorem as we did in Equation 8.3 for Faraday’s law, wewrite ∫

S(∇ ×H) · dS =

∮H · dl =

∫J · dS (8.43)

Applying this relation to the cross-sectional area ABCD of Figure 8.4 we have

H0w = I (8.44)


Front view Left side

E-fieldamplitude

x

x

y

z

z

xJz Jz

–Jz –Jz

Hy

Hy

HyHy

Ex

Ex

ExEx

kz

w

s

y

V

λ

Figure 8.4 Idealised parallel-plate transmission line with no resistive losses inthe current-carrying plates and no shunt conductance between the plates.

where I =∫J · dS, the integral of the current density over ABCD. Now we substi-

tute the definition of inductance L from Equation 8.12 into Equation 8.44, taking the∫B · dS over the surface shown in Figure 8.4. The inductance per unit length along z

is then

dLdz

= L = μ0sw

(8.45)

8.3.2.1 Voltage and current along a transmission line

Equation 8.45 expresses the constant distributed inductance along the planar waveguideof Figure 8.4. The voltage gradient along the guide is then obtained from Equation 8.13:

∂V∂z

= –L∂I∂t

(8.46)

and the current gradient from the distributed capacitance, obtained from Equation 8.17:

∂I∂z

= –C∂V∂t

(8.47)


8.3.3 Correspondence to plane waves

8.3.3.1 Transmission line voltage and current waves

Equations 8.46 and 8.47 bear a close resemblance to Equations 3.18–3.21. Just as wefound for Equations 8.30 and 8.31, differential ‘Helmholtz-like’ equations for V and Iseparately can be obtained for the lossless transmission line following the same procedurethat produced Equations 3.23 and 3.24 for plane waves. Differentiation of Equation 8.46by z and Equation 8.47 by t results in

∂2V∂z2

– LC∂2V∂t2

= 0 (8.48)

and eliminating V in favour of an equation for I by reversing the differentiation variablesfor each equation results in

∂2I∂z2

– LC∂2I∂t2

= 0 (8.49)

Just as in the plane-wave case we associate LC with 1/v2 where v is the phase velocityof the V – I transmission wave running in the two-slab transmission line of Figure 8.4.Furthermore, if we assume time-harmonic phasor solutions to Equations 8.48 and 8.49we can write the solution to the voltage equation as

V = V0ei(βz–ωt) (8.50)

Then substituting this solution into Equation 8.48 yields

∂2V∂z2

+ LCω2V = 0 (8.51)

We see therefore that the phase velocity v and propagation parameter β for the V – Iexcitation of the transmission line is

v =1√LC

and β =ω

v=√LCω (8.52)

Given the voltage solution, Equation 8.50, we can find the current by substitutingEquation 8.50 into Equation 8.46 and integrating with respect to time:

∂I∂t

= –1L∂V∂t

= –1LiβV (8.53)

then

I = –1Liβ∫V dt =

1Lβ

ωV =

√CL· V (8.54)


The ratio of the voltage to current amplitudes has units of resistance and is identifiedwith the characteristic impedance of the transmission line:

Z0 =VI

=

√LC

(8.55)

which is, as expected, in agreement with Equation 8.42.

8.3.3.2 Reflection on a lossless terminated line

So far we have considered the characteristic impedance Z0 of the transmission line itself,which is determined essentially by the geometric parameters of the line— the separ-ation and symmetry between the conductors and the nature of the insulating dielectricbetween them. If now we insert a load impedance ZL across the line conductors, the‘impedance mismatch’ will provoke some fraction of the incident wave to reflect at theload position and the transmission line is said to be ‘terminated’ by ZL . Suppose we putthe load at z = 0. We know that in general V and I on the otherwise lossless line arerelated by

V (z) = V+0 e

iβz + V –0 e

–iβz (8.56)

I(z) =1Z0

[V+

0 eiβz – V –

0 e–iβz] (8.57)

At z = 0 we must have

ZL =V (0)I(0)

=V+

0 + V –0

V+0 – V –

0Z0 (8.58)

so the amplitude of the reflected wave in terms of the line and load impedances is

V –0 =

(ZL – Z0

ZL + Z0

)V+

0 (8.59)

Now, in analogy to Equations 3.139 and 3.140 for plane wave reflection, we define2 areflection coefficient � as

� ≡ –V –

0

V +0

=Z0 – ZLZ0 + ZL

(8.60)

2 We choose here the phase of the reflected voltage to be consistent with the convention used in Section 3.2.3for reflected electromagnetic plane waves. The electrical engineering literature commonly uses � = V –

0 /V+0 .


The voltage and current on the transmission line are then given by

V (z) = V+0

(eiβz – �e–iβz

)(8.61)

I(z) = I+0(eiβz + �e–iβz

)(8.62)

We see that if the load is ‘impedance matched’ to the characteristic impedance of thetransmission line, then reflected waves are suppressed.

Note that ZL is defined as the ratio of voltage to current amplitude at z = 0. Since thesuperposition of incident and reflected waves set up a standing wave on the line, we canexpect the ratio V (z)/I(z) to vary with z. Therefore the impedance along the line varieswith z when the load is mismatched. If we seek the impedance Z a distance –d from theload, then from Equations 8.39, 8.61, and 8.62 we have

Zline = Z0 · e–iβd – �eiβd

e–iβd + �eiβd

= Z0 · (1 – �) – i(1 + �) tanβd(1 + �) – i(1 – �) tanβd

= Z0 · ZL – iZ0 tanβdZ0 – iZL tanβd

(8.63)

Figure 8.5 shows the transmission line terminated in the load ZL and the impedance Zline

a distance –d to the left of the terminating load.

8.3.3.3 Power propagation on a lossless transmission line

The cycle-averaged power on a transmission line is given by the real part of the productof the voltage and current. Using Equations 8.61 and 8.62:

Incident wave Reflected wave

ZL

IL

Zline (−d)

zz = 0

−d

Z0

V +(z), I +(z) V –(z), I –(z) +

−

Figure 8.5 Transmission line impedanceZline(–d) with the line terminated by aload ZL �= Z0. The impedance mismatchsets up a reflection on the line and a stand-ing wave due to the superposition of V+(z)and V –(z) waves.


PT =12Re[V (z)I∗(z)

](8.64)

=12

∣∣V+0

∣∣2Z0

Re[1 + �∗ei2βz – �e–i2βz – |�|2] (8.65)

=12

∣∣V+0

∣∣2Z0

[1 – |�|2] (8.66)

This power expression is reminiscent of the power calculated from the Poynting vectorof a propagating plane wave, Equations 3.166, 3.167, and 3.170. Clearly if the line isimpedance matched so that there is no reflection, � = 0, then all the power propagatesdown the line. If � �= 0 then some power is reflected back along the incident direction,setting up a V – I standing wave along the line.

8.3.3.4 Transmission and reflection at a line junction

Suppose we have two transmission lines, each with a characteristic impedance Z1,Z2,respectively, and they are joined at z = 0 with the Z1 line along z < 0 and the Z2 linealong z > 0. Assume further that the second line with impedance Z2 is infinitely long(or terminated with an impedance matching load) so that there are no reflections at theend of it. At the junction the reflection coefficient is

� =Z1 – Z2

Z1 + Z2(8.67)

so that the voltage along the z < 0 section is

V1(z) = V+0

(eiβ1z – �e–iβ1z

)z < 0 (8.68)

The transmission junction is illustrated in Figure 8.6. On the z > 0 side, the voltagetravels only in the +z direction and the transmitted amplitude is some fraction T of V+

1 .The transmitted wave is then represented by

V2(z) = TV +1 e

iβ2z z > 0 (8.69)

Z1

z

I2

I1

T Z2 V2V1

z = 0

−

+

−

+Γ

Figure 8.6 Two transmission lines joinedat z = 0. The line to the left (z < 0) ischaracterised by line impedance Z1, andthe line to the right (z > 0) by Z2.Reflection coefficient � and transmissioncoefficient T determine the amplitude of thereflected and transmitted waves, respect-ively.


Table 8.1 Equivalent quantities between plane waves and transmission lines.

Plane Wave Transmission Line

Ex(z) V (z)

Hy(z) I(z)

k = ω√με β = ω

√LC

Z =√με Z =

√LC

R �

T T

At the junction itself the two voltages must be equal:

V1(z = 0) = V2(z = 0) (8.70)

V+1 (1 – �) = TV+

1 (8.71)

2Z2

Z1 + Z2= T (8.72)

So we find that in terms of the characteristic impedances in the two lines the reflectionand transmission coefficients are analogous to reflection and transmission of plane wavesat a material surface.

� =Z1 – Z2

Z1 + Z2(8.73)

T =2Z2

Z1 + Z2(8.74)

8.3.4 Equivalence of plane waves and transmission lines

The correspondence between electromagnetic plane wave propagation (Section 3.1,Figure 3.1) through media characterised by μ, ε, and voltage/current transport alonga lossless transmission line (Section 8.3.2) characterised by L,C, is indicated inTable 8.1. Analogous Helmholtz-like wave equations govern propagation, reflection, andtransmission in both cases.

8.4 Special termination cases

In this section we discuss some special cases for ZL in the lossless terminated transmis-sion line, Figure 8.5 in Section 8.3.3.

Special termination cases 169

8.4.1 Shorted transmission line

If we set ZL = 0 then the reflection coefficient, from Equation 8.60, becomes unity,� = 1, and from Equations 8.61 and 8.62 the transmission voltage and current become

V (z) = V +0

(eiβz – e–iβz

)= i2V +

0 sinβz (8.75)

I(z) = I+0(eiβz + e–iβz

)= 2I+0 cosβz (8.76)

Note that at the load point (z = 0) the voltage is null and the current is maximum aswould be expected from a short circuit. Note also that since the product V (z)I∗(z) ispure imaginary no power is delivered to the load. The impedance along the line, fromEquation 8.63 is

Zline = –iZ0 tanβd (8.77)

8.4.2 Open circuit transmission line

Suppose ZL =∞ so that in Figure 8.5 the load is completely removed. In this case, fromEquation 8.60, � = –1, and there is 100% reflection at the load. The voltage and currentexpressions become

V (z) = V +0

(eiβz + e–iβz

)= 2V+

0 cosβz (8.78)

I(z) = I+0(eiβz – e–iβz

)= i2I+0 sinβz (8.79)

As expected, the voltage is maximum at the load point and the current is null. Theexpression for the impedance along the line, from Equation 8.63, is

Zline = iZ0 cotβd (8.80)

and again no power is transmitted to the load.

8.4.3 Line impedance at d = –λ/2

At half-wave points along the line, Equation 8.63 shows that

Zline = ZL (8.81)

At these points the impedance is independent of the transmission line characteristics.


8.4.4 Line impedance at d = –λ/4

At the quarter-wave points we have, from Equation 8.63,

Zline = Z1ZL – iZ1 tanβdZ1 – iZL tanβd

(8.82)

limβd→π /2

Zline =Z21

ZL(8.83)

This quarter-wave transformer can be used as a λ/4 length of line, with impedance Z1 tomatch an input transmission line of impedance Z0 to a given load ZL . Figure 8.7 showshow this matching can be accomplished. In order to suppress reflection at the Z0/Z1

junction, � must be set equal to zero there. Therefore,

Zline = Z0 at – d = λ/4 (8.84)

and

Z1 =√Z0ZL (8.85)

The line impedance at the quarter-wave point, Equation 8.84, is called a quarter-wave transformer because the load impedance is ‘transformed’ by the square of thecharacteristic impedance of the transmission line.

8.4.5 Slightly lossy transmission lines

An ideal, lossless transmission line is characterised by L and C, themselves calculatedfrom Maxwell’s equations and the geometry of the line. However, a real transmis-sion line always has some resistance in the conducting elements and some leakage

Incident wave Reflected wave

ZL

IL

Zline (−d)

zz = 0

−d

Z0

V +(z), I +(z) V –(z), I –(z) +

−

Figure 8.7 Diagram of how a segment oftransmission line with impedance Z1 andlength λ/4 can be used to match an incom-ing transmission line Z0 with a load ZL.The distance -d is equal to a quarterwavelength.

Special termination cases 171

current in the dielectric separating them. As shown in Figure 8.3, these departuresfrom ideality are characterised by R in the conductors and G, the ‘shunt conductance’through the insulating dielectric material. The complex propagation parameter is givenby Equation 8.32,

β2 = (R – iωL)(G – iωC) (8.86)

or

β = ±√(R – iωL)(G – iωC) (8.87)

Assuming that R ωL and G ωC we can write Equation 8.87 as

β = ±iω√LC√1 + i

RωL·√1 + i

GωC

(8.88)

Expanding the square-root expression as the first two terms in a Taylor’s series,

β = ±iω√LC(1 +

i2RωL

)·(1 +

i2GωC

)(8.89)

= ±iω√LC(1 +

i2RωL

+i2GωC

–RG

(ωLC)2

)(8.90)

Since R/ωL and G/ωC are already small compared to unity, the last term on the rightcan be dropped, and we write

β � ±iω√LC[1 +

i2

(GωC

+RωL

)](8.91)

� ±iω√LC ∓ 12

(G

√LC

+ R

√CL

)(8.92)

We then identify the real propagation parameter β:

β = ω√LC (8.93)

and the dissipation term γ :

γ =12

(G

√LC

+ R

√CL

)(8.94)

=12

(GZ0 +

RZ0

)(8.95)


The signs of β and γ have been chosen so that a wave propagating in the positive zdirection has a positive argument, eiβz, and the real exponential term γ dissipates theamplitude to zero as z→∞:

eiβz = ei(β+iγ )z = e(iβ–γ )z (8.96)

8.5 Waveguides

Up to now we have studied plane waves in Chapter 3 and transmission lines in thepresent chapter. Waves propagating within or on guiding structures, which essentiallydefine the transverse field boundary conditions, are related to unbounded plane wavesand are even more closely related to transmission lines. But guided waves also exhibitunique characteristics that require extension of our familiar ideas of voltage, current,capacitance, and inductance. Waveguiding at microwave frequencies was developed forradar during WWII, but now integrated optical circuits at the nanoscale use waveguidesto transport optical signals onto and around integrated microchips. We discuss in thissection the elements of waveguide theory and the relation guided waves bear to planeelectromagnetic waves and to transmission lines.

The prototypical transmission line, consisting of two parallel slab conductors separ-ated by a dielectric, is shown in Figure 8.4. In a sense, this structure can be considered awaveguide since the E-field terminates on the two conducting slabs separated by a con-stant distance in the y – z plane, although the H-field is unbounded along y. It is clearfrom the front and side views, that the E- and H-field lines are confined to the x–y planewith no field components along the direction of propagation, z. Waves with this propertyare called transverse electromagnetic (TEM) waves. An ideal plane wave propagating inspace also exhibits phase fronts confined to the plane orthogonal to the propagation dir-ection, so they are TEM waves as well. Note that if we added two conducting slabs inthe x – z plane so as to enclose a volume with a single conductor of rectangular x – ycross section, no TEM field could exist since the four conducting sides would define aninterior space of constant voltage (and hence a null E-field). However, waveguides withrectangular, cylindrical, and even arbitrarily shaped cross sections do exist, but in orderto understand them we have to extend our ideas beyond the familiar TEM wave. Wavespropagating within structures of a single conductor are classified into two types: trans-verse electric (TE) waves with an H-field component along the propagation direction z(but E-field only in the transverse x–y plane), and transverse magnetic (TM) waves withan E-field component along the propagation direction (but only with transverse H-fieldcomponents). Note that TE and TM waves (or modes) should not be confused with TEand TM polarisation discussed at length in Chapters 3 and 7.

8.5.1 Field solutions for TM and TE waves

We assume field solutions to the Helmholtz equation that will have some spatial vari-ation in the x – y plane, subject to the boundary conditions of a known perfect electric

Waveguides 173

conductor (PEC) guide structure, and propagate along the z direction with eiβz. For thetime being we will take the propagation parameter to be real (lossless medium). Thegeneral form of the phasor fields are then

E(x, y, z) =[E (x, y) + Ez(x, y)z

]eiβz (8.97)

H(x, y, z) =[H (x, y) + Hz(x, y)z

]eiβz (8.98)

Once again we write the two curl equations of Maxwell with harmonic time dependence,no current sources in non-magnetic material (μ = μ0):

∇ × E = iωμ0H

∇ ×H = –iωεE

and write out explicitly the various components ofH and E. For the H-field componentswe have

∂Ez∂y

– iβEy = iωμ0Hx (8.99)

iβEx –∂Ez∂x

= iωμ0Hy (8.100)

∂Ey∂x

–∂Ex∂y

= iωμ0Hz (8.101)

and for the E-field components

∂Hz

∂y– iβHy = –iωεEx (8.102)

iβHx –∂Hz

∂x= –iωεEy (8.103)

∂Hy

∂x–∂Hx

∂y= –iωεEz (8.104)

We can use Equations 8.99 and 8.103 to write Hx in terms of derivatives of Ez and Hz inthe transverse x – y plane. Similarly we can use Equations 8.100 and 8.102 to also writeHy in terms of transverse derivatives of Ez,Hz. Proceeding in the same way for Ex andEy we have finally four equations,


Hx = –i(

k2 – β2) [ωε∂Ez

∂y– β

∂Hz

∂x

](8.105)

Hy =i(

k2 – β2) [ωε∂Ez

∂x+ β

∂Hz

∂y

](8.106)

Ex =i(

k2 – β2) [ωμ0

∂Hz

∂y+ β

∂Ez∂x

](8.107)

Ey = –i(

k2 – β2) [ωμ0

∂Hz

∂x– β

∂Ez∂y

](8.108)

where k, as always, is related to the frequency ω and velocity of light through the medium(v = 1/

√εμ0) by the expression k = ω

√εμ0. The factor

(k2 – β2

)can be interpreted as

the square of a propagation parameter in the x–y plane since β is always along the z-axis.This transverse propagation parameter is called the ‘cutoff ’ parameter kc:

k2c = k2 – β2 (8.109)

8.5.1.1 TE waves

In the case of TE waves, Ez = 0 and Equations 8.105–8.108 become

Hx =ik2c· β ∂Hz

∂x(8.110)

Hy =ik2c· β ∂Hz

∂y(8.111)

Ex =ik2c· ωμ0

∂Hz

∂y(8.112)

Ey = –ik2c· ωμ0

∂Hz

∂x(8.113)

In order to find the transverse fields we have to use the Helmholtz equation to find thepermitted values of Hz. We write the expression

(∇2 + k2)Hz =

(∂2

∂x2+∂2

∂y2+∂2

∂z2+ k2

)Hz = 0 (8.114)

But since

∂2Hz

∂z2= –β2Hz (8.115)

Waveguides 175

the Helmholtz equation reduces to a differential equation for the transverse Hz fieldgradients:

(∂2

∂x2+∂2

∂y2+ k2c

)Hz = 0 (8.116)

Allowed solutions will be a function of the transverse boundary conditions.

8.5.1.2 TM waves

For TM waves we follow the same procedure except we set Hz = 0 and writeEquations 8.105–8.108 as

Hx = –ik2c· ωε∂Ez

∂y(8.117)

Hy =ik2c· ωε∂Ez

∂x(8.118)

Ex =ik2c· β ∂Ez

∂x(8.119)

Ey =ik2c· β ∂Ez

∂y(8.120)

The Helmholtz equation for Ez reduces to(∂2

∂x2+∂2

∂y2+ k2c

)Ez = 0 (8.121)

and allowed solutions will again be a function of the transverse boundary conditions.

8.5.2 Parallel plate waveguide

Here we apply the formal development of the last section to a simple but very useful ex-ample, the parallel plate waveguide. This structure is essentially the same as the parallelplate transmission line studied in Section 8.3.2, but here we will see that the transmissionline TEM waves (with no field components in the direction of propagation) can be com-plemented with TE modes and TMmodes. The two conducting plates of the waveguideare separated by s between the plates (along the x direction) and have a width w (alongthe y direction) that is much greater than the separation s. We can therefore assume thatthe solutions we seek will not be functions of y.

8.5.2.1 TE modes

In order to find the allowed TE modes we start with the reduced Helmholtz equationfor the TE case, Equation 8.116, and write the solutions for Hz. By inspection, we write


the general solution in terms of the real sin and cos functions and refer to the coord-inate system of Figure 8.4. The amplitudes A and B are determined by the boundaryconditions:

Hz(x, y) = A sin kcx + B cos kcx (8.122)

Since we posit that the plates are perfect conductors (and the intervening dielectriclossless), we have for boundary conditions:

Ey(0, y) = 0 and Ey(s, y) = 0 (8.123)

Then from Equation 8.113:

Ey(x, y) = –ik2c· ωμ0

∂Hz

∂x= –

ik2c· ωμ0

∂Hz

∂xeiβz (8.124)

= –iωμ0

kc[A cos kcx – B sin kcx] eiβz (8.125)

The boundary conditions, Equation 8.123, require that A = 0 and puts a condition onthe argument of the sin term, that at x = s the E-field must be null, so

kcs = nπ n = 1, 2, 3 . . . or kc =nπs

n = 1, 2, 3 . . . (8.126)

The boundary conditions impose that only discrete values of kc are permitted (depend-ing on the plate separation s), and the expression for the propagation parameter alongthe z direction becomes

β = ±√k2 – k2c or β = ±

√k2 –

(nπs

)2(8.127)

The solutions for Hz(x, y) are

Hz(x, y) = Bn cosnπsx (8.128)

or

Hz(x, y) = Bn cos(nπsx)eiβz (8.129)

Note that when k > kc, β is positive and the wave propagates along z. When k = kc,then β = 0 and the wave is stationary with zero phase velocity. If k falls below kc, thenβ becomes pure imaginary. The wave no longer propagates in the waveguide but decaysexponentially to 1/e of its initial amplitude at a characteristic distance of x = 1/β. Thepropagating wave becomes an evanescent wave. This threshold behaviour for propagation

Waveguides 177

of mode n within the waveguide is the reason that kc is called the ‘cutoff ’ parameter.Below cutoff, the wave does not propagate. The cutoff parameter kc is related in theusual way to wavelength and frequency:

kc =2πλc

and ωc = kcv (8.130)

where v is the propagation velocity in the gap dielectric of the waveguide. If the dielectricis air or vacuum, then is the velocity of light, c. For a given waveguide ‘mode’ n the cutofffrequency must be greater than kcnv.

Once we have the solutions for Hz(x, y) we can get the solutions for all the transversefield components from Equations 8.110–8.113. Since there is no transverse dependencein the y direction, Equations 8.111 and 8.112 show that Hy = Ex = 0 and

Hx = –iβ

kcBn sin

(nπsx)eiβz (8.131)

Ey = iωμ0

kcBn sin


Note the difference in sign betweenHx and Ey. This sign difference is consistent with thedirection of energy propagation, determined by the Poynting vector, along the positive zdirection, S = E×H. The wave impedance is given by the ratio of the E-field amplitudeto the H-field amplitude,

ZTE =|Ey||Hx| =

ωμ0

β=kβZ0 (8.133)

For the parallel plate waveguide the TE modes consist of a ‘triplet’ of components,Hz,Hx,Ey and the modes are labelled TEn, n = 1, 2, 3 . . . .

8.5.2.2 TM modes

To find the field components for the TM modes we follow a parallel procedure to theTE case. The first step is to find the Ez solutions from the reduced Helmholtz equation,Equation 8.121, subject to the boundary conditions on the parallel plate waveguide atx = 0 and x = s. Once again the general solution is

Ez(x, y) = A sin kcx + B cos kcx (8.134)

with boundary conditions that Ez(0, y) = Ez(s, y) = 0. Therefore B = 0 and sin kcx = 0when x = 0 or when x = s. The solutions to Equation 8.121 that respect the boundaryconditions are therefore

Ez = An sin(nπs

)x n = 0, 1, 2, 3 . . . (8.135)


and

Ez = An sin(nπs

)xeiβz (8.136)

with

β = ±√k2 – k2c = ±

√k2 –

(nπs

)2(8.137)

From Equations 8.117–8.120 we find the transverse field components:

Hy = iωε

kcAn cos


Ex = iβ

kcAn cos


The impedance for the TM waves is

ZTM =ExHy

=β

ωε=β

kZ0 (8.140)

and the TM modes consist of a triplet of three field components: Ez,Ex,Hy. Note thatfor the lowest n = 0 mode, there is no cut-off frequency, Ez = 0, and the Hy,Ex compo-nents have constant amplitudes. In fact the TM0 mode is identical to the TEM0 mode,represented in Figure 8.4.

8.6 Rectangular waveguides

The parallel plate waveguide is essentially a one-dimensional problem since the trans-verse spatial variation of the TE and TM modes depend only on x. Here we examinethe more realistic case of a rectangular waveguide with finite dimensions in both x andy. We assume again boundary conditions on a perfect electrical conductor (PEC) andapply those E-field conditions at the waveguide edges along x and y. By convention wedenote the long and short sides of the rectangular cross section as l, and h, respectively,and align l along x and h along y so that the lower left corner of the waveguide crosssection is at the coordinate origin. The guide is filled with some non-magnetic, losslessdielectric material characterised by permeability μ0 and permittivity ε. Figure 8.8 showsthe layout for the rectangular waveguide.

8.6.1 TE modes

The transverse electric (TE) modes are defined as those waves that propagate in theguide with E-fields only transverse to the z (propagation) direction. Therefore, we seekHz solutions to the transverse Helmholtz equation:

Rectangular waveguides 179

2b

2ax

y

z

Figure 8.8 Rectangular waveguide with sides x = 2a and y = 2b.Inside the conducting boundaries the waveguide is filled with alossless dielectric characterised by μ0, ε.

(∂2

∂x2+∂2

∂y2+ k2c

)Hz(x, y) = 0 (8.141)

The solutions will be functions of both coordinates, but we can expect them to beuncoupled. Therefore we can posit solutions of the form Hz(x, y) = X(x)Y(y).Substitution into Equation 8.141 and division by X(x)Y(y) results in

1X∂2X∂x2

+1Y∂2Y∂y2

= –k2c (8.142)

Since this result must be true for all values of x, y, the terms on the left must themselvesbe equal to constants that we write

1X∂2X∂x2

= –k2x1Y∂2Y∂y2

= –k2y (8.143)

and

k2x + k2y = k2c (8.144)

The solution for Hz(x, y) is a product of the general solution in x and the general solutionin y:

Hz(x, y) = (A cos kxx + B sin kxx) ·(C cos kyy +D sin kyy

)(8.145)


The boundary conditions at x, y = 0 and x = a, y = b are

Ex(x, 0),Ex(x, b) = 0 and Ey(0, y),Ey(a, y) = 0 (8.146)

The transverse E-fields are obtained from the longitudinal H-field Hz by usingEquations 8.112 and 8.113:

Ex = iωμ0

k2cky (A cos kxx + B sin kxx) ·

(–C sin kyy +D cos kyy

)(8.147)

Ey = iωμ0

k2ckx (–A sin kxx + B cos kxx) ·

(C cos kyy +D sin kyy

)(8.148)

The condition on Ex(x, y = 0) implies that D = 0 and the condition on Ex(x, y = b)implies that ky = nπ /b with n = 0, 1, 2 . . .. Similarly, the condition on Ey(x = 0, y)requires that B = 0 and the condition on Ey(x = a, y) requires kx = mπ /a with m =0, 1, 2 . . . . Therefore, the general solution for Hz(x, y), Equation 8.145, becomes

Hz = Amn cosmπxa

cosnπyb

(8.149)

and

Hz(x, y, z) = Amn cosmπxa

cosnπybeiβz (8.150)

The arbitrary amplitude constant Amn is a product of the constants A,C in Equa-tion 8.145. Now we get the transverse fields using Equations 8.110–8.113:

Hx = –iβmπk2c a

Amn sinmπxa


Hy = –iβnπk2c b

Amn cosmπxa

sinnπybeiβz (8.152)

Ex = –iωμ0nπk2c b

Amn cosmπxa


Ey = iωμ0mπk2c a

Amn sinmπxa


As always we get the cutoff propagation parameters and frequencies from

β =√k2 – k2c =

√k2 –

(k2x + k2y

)=

√k2 –

(mπa

)2–(nπb

)2(8.155)

Rectangular waveguides 181

The propagation parameter along z becomes real when

k > kc =

√(mπa

)2–(nπb

)2(8.156)

The TE wave impedance is denoted by ZTE and is independent of the mode label-ling n,m:

ZTE =ExHy

=kβZ0 (8.157)

Because we have posited that a > b, Equation 8.156 shows that the TE mode with thelowest cutoff frequency is TEmn = TE10.

8.6.2 TM modes

The transverse magnetic modes are defined as those modes with H-fields only in thetransverse (x – y) plane. Therefore, the field component in the z direction must bean E-field. The waves propagate along z and Ez is written as the product Ez(x, y)eiβz.Therefore, we seek Ez solutions to the same Helmholtz wave equation as for the TEmodes: (

∂2

∂x2+∂2

∂y2+∂2

∂z2+ k2c

)Ez = 0 (8.158)

The general solution for Ez has the same form as that for Hz, Equation 8.145, andsubject to boundary conditions,

Ez(x, 0),Ez(x, b) = 0 and Ez(0, y),Ez(a, y) = 0 (8.159)

The expression for Ez is therefore

Ez(x, y, z) = Ez(x, y)eiβz = Bmn sinmπxa


with m, n = 1, 2, 3 . . . . As with the TE modes the cutoff propagation parameter isgiven by

k2c = k2x + k2y (8.161)

and the boundary conditions dictate that

kx =mπa

and ky =nπb

(8.162)


The transverse field components are obtained from Ez by using Equations 8.117–8.120:

Hx = –iωεnπbk2c

Bmn sinmπxa


Hy = iωεmπak2c

Bmn cosmπxa


Ex = iβmπak2c

Bmn cosmπxa


Ey = iβnπbk2c

Bmn sinmπxa


The propagation parameter is given, as usual, by

β = ±√k2 – k2c = ±

√k2 –

(mπa

)2–(nπb

)2(8.167)

We can see from Equations 8.163–8.166 that TM01 or TM10 vanish (in fact all TMmodes with a zero index vanish) so the lowest TM mode is TM11 with a cutoffpropagation parameter given by

kc =

√(πa

)2+(πb

)2(8.168)

If one of the sides becomes much longer than the other (say, a� b), then

kc→ π

b(8.169)

and the cutoff kc becomes equal to the parallel plate waveguide case (Section 8.5).Finally, the TM mode impedance is given by

ZTM =ExHy

=β

kZ0 (8.170)

8.7 Cylindrical waveguides

The cylindrical geometry is important not only for waveguides and transmission lines inthe conventional microwave domain but also for nanoscale holes and hole arrays in thevisible and near infrared regions. A sensible point of departure for understanding lighttransmission through these cylindrically symmetric structures is an analysis of the fieldmodes they can support. The procedure is really no different than what we have donefor the rectangular waveguides. We choose a coordinate system in which the solutions tothe Helmholtz equation will separate, and we will be able to write the solutions to Ez and

Cylindrical waveguides 183

y

r

x

a

z

φ

Figure 8.9 Schematic of cylindrical wave-guide. The cylinder surface is assumed to bea perfect conductor. Cylindrical coordinatesare r,ϕ and z as shown. The radius of thecylinder is a.

Hz as a product of three factors R(r)(ϕ)eiβz where R(r) is the radial field dependenceand (ϕ) is the azimuthal angular dependence around the z-axis. A schematic of theguide is shown in Figure 8.9.

8.7.1 TE modes

As always, TE modes are defined by the presence of electric fields only in the transverseplane. The field function along the z-axis must be an H-field, and we write the Helmholtzequation.

∇2Hz + k2Hz = 0 (8.171)

Then we write the ∇2 or Laplacian operator in cylindrical coordinates and factorHz(r,ϕ, z) = Hz(r,ϕ)eiβz. We take the expression for the Laplacian operator fromEquation D.32 of Appendix D:

(∂2

∂r2+

1r∂

∂r+

1r2∂2

∂ϕ2+ k2c

)Hz(r,ϕ) = 0 (8.172)

Now we take the posited product solution

Hz(r,ϕ) = R(r)(ϕ) (8.173)

and substitute it into Equation 8.172. After multiplication by r2 we have

r2

Rd2Rdr2

+rRdRdr

+ r2k2c = –1

d2dϕ2

(8.174)


The left-hand side is a function only of r and the right-hand side only of ϕ. In orderfor the equation to be valid for the entire range of r,ϕ the two sides must be equal to aconstant. We set the separation constant equal to l2ϕ and write

–1

d2dϕ2

= l2ϕ (8.175)

or rearranging:

d2dϕ2

+ l2ϕ = 0 (8.176)

From the left-hand side of Equation 8.174 we have

r2d2Rdr2

+ rdRdr

+(r2k2c – l

2ϕ

)R = 0 (8.177)

Note that lϕ is unitless. From inspection we can write down a solution to Equation 8.176:

= eilϕϕ (8.178)

Now in order to represent a physical entity must be a single-valued function, meaningthat [lϕ(ϕ + 2π)] must have the same value as (lϕϕ) or

[(lϕ(ϕ + 2π)] = eilϕϕeilϕ2π = (lϕϕ) (8.179)

Therefore, the separation constant lϕ must be an integer, lϕ = n = 1, 2, 3 . . . . The generalsolution to Equation 8.176 can also be written in terms of real sin and cos functions:

(ϕ) = A sin nϕ + B cos nϕ (8.180)

The radial equation becomes

r2d2Rdr2

+ rdRdr

+(r2k2c – n

2)R = 0 (8.181)

This expression is one of the forms of Bessel’s differential equation. There exists a fam-ily of functions that are solutions to this equation. In fact there are two independentfamilies, Bessel functions of the first kind, Jn(x), and of the second kind, Nn(x). Thefunctions of the second kind, the Neumann functions, tend to –∞ as x → 0 so theyare not physically acceptable solutions. The Bessel functions of the first kind are finiteat the origin and exhibit physically acceptable behaviour. The first few Bessel functions


1.00

J0(x)

J1(x)

J2(x)

0.85

0.70

0.55

0.40

0.25

0.10

–0.05

–0.20

–0.35

–0.500 2 4 6 8 10

x12 14 16 18 20

Figure 8.10 First three Bessel functions Jn(x).

Jn(x) are plotted in Figure 8.10. Putting together the angular and radial solutions to thereduced Helmholtz equation, we have

Hz(r,ϕ) = (A sin nϕ + B cos nϕ) Jn(kcr) (8.182)

The next step is to apply boundary conditions at r = 0 and r = a, where a is the radiusof the hole. But in order to do so we must write the E-field and H-field components inthe r,ϕ directions in terms of the z-axis components. These components are found fromwriting down the six Maxwell curl equations in cylindrical coordinates and reducingthem to four relations of r,ϕ in terms of z as was done for Equations 8.105–8.108. Theresult is

Er = i1k2c

(β∂Ez∂r

+ωμ0

r∂Hz

∂ϕ

)(8.183)

Eϕ = i1k2c

(β

r∂Ez∂ϕ

– ωμ0∂Hz

∂r

)(8.184)

Hr = –i1k2c

(ωε

r∂Ez∂ϕ

– β∂Hz

∂r

)(8.185)

Hϕ = iik2c

(ωε∂Ez∂r

+β

r∂Hz

∂ϕ

)(8.186)

On the perfectly conducting wall of the cylinder, the E-field tangent to the wall mustvanish. Therefore, Eϕ(r = a,ϕ) = 0. From Equation 8.184 we have

Eϕ(r,ϕ, z) = –iωμ0

kc(A sin nϕ + B cos nϕ) J′n(kcr)e

iβz (8.187)


where J′n(kcr) is the radial derivative of the Bessel function. Applying the boundarycondition at r = a:

Eϕ(a,ϕ, z) = –iωμ0

kc(A sin nϕ + B cos nϕ) J′n(kca)e

iβz = 0 (8.188)

Since the r dependence is only in the Bessel term, we must have

J′n(kca) = 0 (8.189)

But this requirement fixes kc to the zeros of J′n(kca). Each function J′n, labelled by n, willhave a series of zeros where J′n [(kca)nm] corresponds to the mth zero of the nth Besselfunction derivative. The arguments of the zeros (or roots) of J′n are not simple multiplesof π as in the case of sin and cos functions, they have to be found numerically and Table8.2 provides values for the first few arguments (kca)nm corresponding to the zeros ofthe J′n functions. The first three Bessel function derivatives are plotted in Figure 8.11.Evidently we obtain kc for the nmth mode from

kcnm =kcanma

(8.190)

We obtain the propagation parameter along z for the nmth mode in the usual way,

βnm =√k2 – k2cnm (8.191)

0.6

0.5J1

´(x)

J2´(x)

J0´(x)

0.4

0.3

0.2

0.1

0

–0.1

–0.2

–0.3

–0.4

–0.5

–0.60 2 4 6 8 10

x12 14 16 18 20

Figure 8.11 Derivatives of the first three Bessel functions, showing thepositions of the first few zero-crossings.


Table 8.2 Some values of (kca)nm for TE in a cylindrical waveguide.

n (kca)n1 (kca)n2 (kca)n3

0 3.832 7.016 10.174

1 1.841 5.331 8.536

2 3.054 6.706 9.970

and with βnm in hand we construct Hz(r,ϕ, z) = Hz(r,ϕ)eiβnmz. Then from Equa-tions 8.183–8.186 we calculate the transverse fields for the nmth mode. The notationnm has been omitted from kcnm and βnm so as not to encumber the expressions morethan necessary:

Er = iωμ0nk2c r

(A cos nϕ – B sin nϕ) Jn(kcr)eiβz (8.192)

Eϕ = –iωμ0


iβz (8.193)

Hr = iβ


iβz (8.194)

Hϕ = iβnk2c r

(A cos nϕ – B sin nϕ) Jn(kcr)eiβz (8.195)

In fact, the A and B amplitude factors in the angular part of the transverse fieldexpressions are not independent. We can easily see this from Equation 8.178. The‘normalisation integral’ of the angular solution in complex form is

∫ 2π

0∗ dϕ = 2π = const. (8.196)

and writing the normalisation of the angular solution in terms of real sin and cosfunctions,

∫ 2π

0(A sin nϕ + B cos nϕ)2 dϕ =

12

(A2 + B2) = const. (8.197)

or in other words the only restriction on the amplitudes of the sin and cos terms is thatthe sum of the squares of A and B must always be equal to some constant. Therefore,


we can set A = 0 and let B2 = const. Then the transverse fields can be written in asomewhat simpler form:

Er = –iωμ0nk2c r

(B sin nϕ) Jn(kcr)eiβz (8.198)

Eϕ = –iωμ0

kc(B cos nϕ) J′n(kcr)e

iβz (8.199)

Hr = iβ


iβz (8.200)

Hϕ = –iβnk2c r


The TE mode impedance is given by

ZTE =ErHϕ

=kβZ0 (8.202)

8.7.2 TM modes

In the case of TM modes, there is no component Hz along the axis of symmetry, and weseek solutions to the Helmholtz equation for Ez(r,ϕ, z) = Ez(r,ϕ)eiβz. As with the TEmodes we have a reduced Helmholtz equation for the transverse coordinates:(

∂2

∂r2+

1r∂

∂r+

1r2∂2

∂ϕ2+ k2c

)Ez(r,ϕ) = 0 (8.203)

This expression is the same as Equation 8.172 so the general solutions are the same:

Ez(r,ϕ) = (A sin nϕ + B cos nϕ) Jn(kcr) (8.204)

But in this case the boundary conditions on the E-field can be applied directly to thesolutions for the Helmholtz equation:

Ez(r = a,ϕ) = 0 (8.205)

Therefore, from Equation 8.204 we have

Jn(kca) = 0 (8.206)

and we get the cutoff parameter for the TMnm mode from the zeros of Jn Bessel function.Table 8.3 shows some values for the roots of Jn. Once again, the cutoff parameters areobtained from

kcnm =kcanma

(8.207)

Networks of transmission lines and waveguides 189

Table 8.3 Some values of (kca)nm for TM in a cylindricalwaveguide.

n (kca)n1 (kca)n2 (kca)n3

0 2.405 5.520 8.654

1 3.832 7.016 10.174

2 5.135 8.417 11.620

The propagation parameter along z is

βnm =√k2 – k2c (8.208)

The transverse fields from Equations 8.183–8.186 and setting A = 0 are

Er = iβ


iβz (8.209)

Eϕ = –iβnk2c r


Hr = iωεnk2c r


Hϕ = iωε


iβz (8.212)

Finally, the mode impedance is given by

ZTM =ErHϕ

=β

kZ0 (8.213)

8.8 Networks of transmission lines and waveguides

Earlier in this chapter we discussed the properties of lumped circuit elements and howthese properties, such as capacitance, inductance, and resistance, can be adapted to dis-tributed structures like transmission lines and waveguides. The purpose of the presentsection is to show how transmission lines and waveguides themselves can be groupedtogether into a network to achieve some predetermined design result without having tosolveMaxwell’s equations directly for elaborate source and boundary conditions. The ul-timate goal is then to adopt these ideas to nanoscale structures in order to build networksof 2-D surface or gap plasmon waveguides, or 1D interfacial waveguides. The discussionhere will be somewhat brief since an extended discussion of networked-transmission-linetheory is really a digression from the subject of light–matter interaction. The idea is tointroduce key concepts and notation so that readers not trained in electrical engineeringcan grasp the significance of these analysis tools.


8.8.1 Impedance and admittance matrices

Figure 8.12 shows a generalised waveguide network where we see propagation into andaway from some arrangement of couplers, dividers, and amplifiers, etc. that constitutethe network. Often our interest in focused on the inputs and outputs at the network‘ports’, labelled in the figure as ports S1 – S4. The ports are assumed to be single-modewaveguides, and we can in principle, know the transverse wave properties and the propa-gation parameter of the active input and output ports. We can write the voltage andcurrent passing the nth port entrance-exit plane:

Vn = V +n + V –

n

In = I+n – I–n

and relating the voltages and currents by the impedance Z for each port of the four-portnetwork of Figure 8.12,

⎡⎢⎢⎣V1

V2

V3

V4

⎤⎥⎥⎦ =

⎡⎢⎢⎣Z11 Z12 Z13 Z14

Z21 Z22 Z23 Z24

Z31 Z32 Z33 Z34

Z41 Z42 Z43 Z44

⎤⎥⎥⎦ ·

⎡⎢⎢⎣I1I2I3I4

⎤⎥⎥⎦ (8.214)

NetworkV +1, I +1

V –1, –I –1

V –4, –I –4V –3, –I –3

V –2, –I –2V +2, I +2

S1

S4

S3

S2

Figure 8.12 A schematic waveguide network with two ports S1 andS2 showing inputs and outputs and two ports S3 and S4 showingonly outputs. The outputs at S1 and S2 can be thought of as reflectedwaves.


or in short-hand matrix notation,

[V ] = [Z] · [I ] (8.215)

The inverse of the impedance is called the admittance Y , and the inverse of theimpedance matrix [Z] is the admittance matrix [Y ]:

[Y ] = [Z]–1 (8.216)

or written out explicitly for the four-port case, Equation 8.216 becomes

⎡⎢⎢⎣I1I2I3I4

⎤⎥⎥⎦ =

⎡⎢⎢⎣Y11 Y12 Y13 Y14

Y21 Y22 Y23 Y24

Y31 Y32 Y33 Y34

Y41 Y42 Y43 Y44

⎤⎥⎥⎦ ·

⎡⎢⎢⎣V1

V2

V3

V4

⎤⎥⎥⎦ (8.217)

The matrix equation, Equation 8.215, means that for each element Zij we have

Zij =ViIj

(8.218)

where all Ik with k �= j have been set to zero. Equation 8.218 means that we can determinethe impedance Zij if we inject current I at port j and measure the resulting voltage V atport i with all the other ports closed or ‘open-circuited’ in engineering parlance. For theadmittance matrix we have the analogous situation:

Yij =IiVj

(8.219)

which means that in order to determine the admittance matrix element Yij , we apply avoltage V to port j and measure the current I at port i with all other ports disconnectedfrom the network (open-circuited). It is important to remember that the network canbe lossy, in which case the matrix elements Zij or Yij are complex. It can be shown thatlossless networks have all Zij and Yij pure imaginary and that reciprocal networks arerepresented by [Z] and [Y ] matrices that are symmetric with respect to an interchangeof indices. That is, for example, Zij = Zji. Physically, the reciprocal property meansthat if a current injected at port m produces a voltage at port n, then injecting the samecurrent in port n will produce the same voltage at port m. Networks of linear, passivecomponents, such as resistors, capacitors, and inductors, are reciprocal. Networks withactive non-linear components with gain, such as amplifiers, are not reciprocal.


8.8.2 Illustrative example: two-port voltage divider

We illustrate these ideas with a simple two-port voltage divider network depicted inFigure 8.13. The first step is to shut off port S2 and inject current I1 at port S1. Then:

Z11 =V1

I1= ZA + ZC (8.220)

because no current runs through ZB. The ‘transfer impedance’ Z12 is obtained by shut-ting off port S1 and injecting current into port S2. No current flows through ZA so all theinjected current I2 flows through ZB and ZC. Then V2 = I2(ZB + ZC). The voltage V1

appearing at the port S1 is just the current I2 flowing through the ‘pull-up’ resistor ZC.Therefore, Z12 is ZC. It is easy to verify that this network is reciprocal and that, therefore,Z22 = Z11 and Z12 = Z21.

8.8.3 The scattering matrix

For high-frequency networks in the microwave and optical regime, it is difficult to char-acterise voltage and current points because the transverse amplitudes across wavefrontsneed not be constant. Therefore, instead of using impedance and admittance, the scat-tering matrix (or S-matrix) is often employed instead. An S-matrix relates voltage wavesincident and reflected at the various network ports. Incident and reflected power fromtravelling and standing waves lend themselves to easier measurement in high-frequencynetworks. Consider port 1 in Figure 8.12. The S-matrix relating the incident voltagewave amplitude V +

1 to the reflected wave amplitude V –1 is given by

⎡⎢⎢⎣V –

1V –

2V –

3V –

4

⎤⎥⎥⎦ =

⎡⎢⎢⎣S11 S12 S13 S14

S21 S22 S23 S24

S31 S32 S33 S34

S41 S42 S43 S44

⎤⎥⎥⎦ ·

⎡⎢⎢⎣V +

1V +

2V +

3V +

4

⎤⎥⎥⎦ (8.221)

I1

l1 l2

I2

V1 V2

Z

ZA ZB

ZC V2V1

S1 S2

Port 1

++

––

(a)

(b)

Port 1 Port 2

Port 2

Figure 8.13 (a) Two-port impedance network. (b) Two-port impedance network implementing a voltage divider.Note that the directions of I1 and I2 point into the portstowards the network.


The usual rules of matrix multiplication obtain here. Thus, for a given network, thevoltage amplitude of a reflected wave at port i is related to the voltage amplitude of anincident wave at port j by

V –i = SijV +

j (8.222)

with the implicit assumption that all other ports are quiescent with voltages at the portinputs set to zero. The diagonal matrix elements express the ratio of reflected ampli-tude to incident amplitude at a given port, assuming all other ports are quiescent (i.e.matched impedances at all other ports). Thus, Sii is a reflection coefficient at port i.Since the S-matrix is essentially intended for waveguide networks, the ports are assumedto have some characteristic impedance (for example, Z0 = 50�). The Sij matrix elementexpresses the transmission coefficient from port j to port i.

8.8.4 Relation of S-matrix to Z-matrix

We know that the voltage and current at port n is the net result of transmission andreflection:

Vn = V+n + V –

n (8.223)

In = I+n – I –n (8.224)

Now suppose that the characteristic impedance at port n is unity, Z0n = 1. Then we canalso write

In = I+n – I –n = V +n – V –

n (8.225)

and the voltage, current, and impedance matrices are related by

[V ] = [Z] · [I ] = [Z] · ([V +] – [V –])= [V+] + [V –]

Rearranging by grouping matrices [V +] and [V –] we have

[V+] – [V+][Z] = –[V –] – [Z][V –] = – ([V –] + [Z][V –]) (8.226)

We can matrix-factor this last matrix equation if we define a unit matrix [U ] with onlyelements of unity on the diagonal and zero elements elsewhere. Then we have

[V+] ([U ] – [Z]) = –[V –] ([U ] + [Z]) (8.227)


or

[V+] ([Z] – [U ]) = [V –] ([U ] + [Z])

[V –][V +]

=[Z] – [U ][Z] + [U ]

(8.228)

Now division by a matrix really means multiplication by the matrix inverse so

[V –][V+]

= ([Z] – [U ]) · ([Z] + [U ])–1 (8.229)

and from the definition of the S-matrix, Equation 8.221, we can identify the S-matrixwith the right-hand side of Equation 8.229:

[S] =[V –][V +]

= ([Z] – [U ]) · ([Z] + [U ])–1 (8.230)

8.8.5 The ABCD matrix

Although the preceding sections have assumed that a network has n ports, in most prac-tical cases circuits are designed with many two-port networks in series (in ‘cascade’ inengineering parlance). A given two-port network is described by the matrix equation

[V1

I1

]=[A BC D

]·[V2

I2

](8.231)

In equation form:

V1 = AV2 + BI2 (8.232)

I1 = CV2 +DI2 (8.233)

The diagram in Figure 8.14 shows how the ABCD matrix functions. If we have twonetworks in series, the overall result is obtained by matrix multiplication:

[V1

I1

]=[A1 B1

C1 D1

]·[A2 B2

C2 D2

]·[V3

I3

](8.234)

I1 I2

ABCDnetwork

V1 V2

+

–

+

–

Figure 8.14 Schematic ABCD network. Note thatthe positive current I2 is pointing away from the portentrance. This ABCD-matrix convention differs fromthe S-matrix where all positive quantities point intothe port.


I1 I2Z+

––

+

V1 V2 Figure 8.15 Two-port ABCD network: one imped-ance element.

Y

–

+

–

+

V1 V2

I1 I2

Figure 8.16 Two-port ABCD network: one admit-tance element.

I1 I2Z1 Z2

Z3

++

––

V1 V2

I3 Figure 8.17 Two-port ABCD impedanceT-network.

The ABCD matrices can be used to construct elaborate networks from a few simpleelements. For example, suppose we have an elementary two-port network consisting ofone impedance element, as shown in Figure 8.15. By applying Ohm’s law we can identifythe ABCD matrix elements. By inspection, it is easy to determine that A = 1, B =Z, C = 0, and D = 1. Another example is a two-port network with a single admittanceas shown in Figure 8.16. Again, by Ohm’s law and remembering the Y = 1/Z, a simpleinspection of the coupled equations reveals that A = 1,B = 0,C = Y , and D = 1.

Figure 8.17 shows an impedance T-network. The ABCD matrix for this network canbe obtained by a more systematic approach. First consider the A element and Equa-tion 8.232. We see that if no current flows at the output (open circuit with I2 = 0),then A = V1/V2. But in that case, from Ohm’s law V1 = I1 (Z1 + Z3) and V2 = I1Z1.Therefore, A = (Z1 + Z3) /Z1. To determine the B element we set V2 = 0 (output shortcircuit). Then B = V1/I2. The result of applying Ohm’s law (or the Kirchhoff voltage andcurrent rules) is that B = Z1 +Z2 +Z1Z2/Z3. Similarly, we consider Equation 8.233 andset successively I2,V2 = 0 to find the matrix elements C and D. The result is C = 1/Z3

and D = 1 + Z2/Z3.Another common network is the admittance network shown in Figure 8.18. This

arrangement is often called the Pi-network. By methodically ‘open-circuiting’ and‘shorting’ the output terminals, the elements of the ABCD matrix can be determined.


V1 Y1 Y2

Y3

V2

I1 I2

Figure 8.18 Two-port ABCD admittancePi-network.

I1 I2

V1

Z

Z0, β V2

Figure 8.19 Two-port ABCD network element: transmissionline segment of length z.

N:1

V1 V2

I1 I2

Figure 8.20 Two-port ABCD network element:inductive coupler.

Two more elementary functions useful for constructing more elaborate networks arethe transmission line segment and the inductive transformer. Figures 8.19 and 8.20sketch out these two circuit functions, and the ABCD matrix elements for all theseelementary circuit functions are entered in Table 8.4.

Table 8.4 ABCD elements for frequently used network functions.

Network A B C D

Single Z 1 Z 0 1Single Y 1 0 Y 1T-Impedance 1 + Z1

Z3Z1 + Z2 + Z1Z2

Z31Z3

1 + Z2Z3

Pi-Admittance 1 + Y2Y3

1Y3

Y1 +Y2 + Y1Y2Y3

1 + Y1Y3

Ideal Transmission Line cosβz –iZ0 sinβz – iZ0

sinβz cosβz

Real Transmission Line coshβz Z0 sinhβz1Z0

sinhβz coshβz

Inductive Coupler N 0 0 1N


8.8.6 Reciprocal and lossless networks

Most common network elements such as capacitors, inductors, and resistors are lin-ear, and passive in the sense that they do not contribute to voltage gain. The matrixelements representing these networks are in general complex quantities. Therefore, ingeneral, a Z, Y, S, or ABCD matrix will contain 2N2 independent elements, where Nis the number of ports in the network. For components with linear, passive response,however, the matrices are always symmetric so that Xij = Xji. All the network matrixrepresentations considered here are symmetric. Furthermore, it can also be shown thatif the network is lossless, containing only reactive elements that store power but donot dissipate it, the network elements will be pure imaginary quantities. Coupling net-works that contain capacitance and induction but no significant resistance fall into thiscategory.

8.8.7 Comparison of impedance matrix and ABCD matrix

We can specialise the impedance matrix, discussed in Section 8.8.1, to the two-portnetwork,

[V1

V2

]=

[Z11 Z12

Z21 Z22

]·[I1–I2

](8.235)

Note that the sign of I2 has been reversed. The reason is that we seek to comparethe ABCD network scheme with the Z scheme. But the impedance network shown inFigure 8.13 assumes that positive current I2 points into port 2, whereas in the ABCDnetwork I2 points outwards from the port. Written out in equation form we have

V1 = Z11I1 – Z12I2 (8.236)

V2 = Z21I1 – Z22I2 (8.237)

The set of Z-coupled equations, Equations 8.236 and 8.237, must be compared to theset of ABCD-coupled equations, Equations 8.232 and 8.233. Suppose we take I2 outputcurrent to zero (open-circuit port 2). Then from Equation 8.232 A = V1/V2 and fromEquations 8.236 and 8.237 Z11 = V1/I1 and Z21 = V2/I1. From these relations we findthat the A element of the ABCD matrix is related to the impedance matrix elementsby A = Z11/Z21. If we short-circuit port 2 (V2 = 0) we find B = Z11(Z22/Z21) – Z12.Proceeding similarly for C and D, we find C = 1/Z21 and D = Z22/Z21.

The equivalence between the ABCD matrix elements, the impedance matrix elem-ents, and the impedance-equivalent circuit for the T-network is summarised in Table 8.5.The T-network, implemented with impedance elements, is shown in Figure 8.21.


Table 8.5 Equivalence between ABCD matrix and Z-matrix elements intwo-port networks.

ABCD Matrix Z-Matrix T-Network

AZ11Z21

1 +Z1Z3

BZ11Z22 – Z12Z21

Z21Z1 + Z2 +

Z1Z2Z3

C1Z21

1Z3

DZ22Z21

1 +Z2Z3

Z1 = Z11 – Z21 Z2 = Z22 – Z21

Z3 = Z21

+

–

+

–

I1 I2

V1 V2

Figure 8.21 T-Network equivalent circuit showing Z1,Z2,Z3impedances in terms of Z-matrix elements.

8.9 Nanostructures and equivalent circuits

8.9.1 Nanosphere driven by a harmonic electric field

We discuss how the response of nanoscale objects to light in the visible-near IR rangecan be considered as an ‘equivalent circuit’. Our discussion follows the treatment of N.Engheta and his research group (Reference 3. listed in Section 8.12). We begin byrevisiting the sphere subject to an electric field that we considered in Chapter 2, Sec-tion 2.6. Let us consider a homogenous sphere of nanoscale dimension, radius r0,immersed in a harmonically varying E-field, E = E0e–iωt. We posit the permittivity ofthe material as ε without specifying, for the present, whether the sphere is dielectric orconductive. Since the sphere is subwavelength in size, we can use the quasistatic ap-proximation for the E-fields in the space within and outside the sphere. We have alreadyobtained these fields in Equations 2.121 and 2.122, but we rewrite them here for con-venience. The permittivity of the sphere itself is labelled εin and outside the sphere thepermittivity is εout:

Nanostructures and equivalent circuits 199

y

z

DdipoleE0

Figure 8.22 Polarised sphere subject to an external E-fieldEz = E0e

–iωt . E-field lines are sketched in the vicinity of thesphere, and displacement field (D-field) lines are shown within thesphere and looping outside the sphere in a dipole pattern. Note thatthe D-field lines normal to the surface are continuous through theinterface.

Ein(r, θ) =[

3εoutεin + 2εout

]E0ez (8.238)

Eout(r, θ) = E0ez + E0


]r30r3(2 cos θ er + sin θ eθ

)(8.239)

Figure 8.22 shows the shape and direction of the applied E-field lines (along the zdirection) in the vicinity of the sphere, and the displacement field inside the sphere,Din = εinEin, and outside the sphere, Dout = εoutEout. At the surface of the sphere,matching conditions impose that the normal component of the displacement field becontinuous across the surface:

εinEin · n = εout

[E0ez · n + E0


](2 cos θ er · n + sin θ eθ · n)

]

εin

[3εout

εin + 2εout

]E0ez · n = εout

[E0ez · n + E0


]2ez · n

](8.240)

The applied displacement field, parallel to z, inside the sphere is given by

Dinappl = εinEin = εinE0ez (8.241)

The net or residual displacement field within the sphere is the difference between thetotal displacement field (left-hand side of Equation 8.240) and the applied displacementfield:


Dnet = εin

[εout – εinεin + 2εout

]E0ez (8.242)

The component normal to the sphere surface of this net displacement field is

Dnet · n = εin


]E0ez · n (8.243)

Now we subtract Equation 8.243 from both sides of Equation 8.240 and rearrange theterms to put the applied fields inside and outside the sphere on the left and everythingelse on the right:

(εin – εout)E0ez · n = εin


]E0ez · n + εout


]E02ez · n (8.244)

The displacement current is given by the time derivative of the displacement field.Taking the time derivative of Equation 8.244 and integrating over the surface of thehalf-sphere with positive charge in Figure 8.22 yields

–iω (εin – εout)E0 · πr20 = –iωεin


]E0 · πr20

– iωεout


]E0 · 2πr20 (8.245)

The integration over the surface is carried out by noting that the ez · n factor in Equa-tion 8.244 is cos θ and that therefore the integration over an infinitesimal of half thesphere surface dS is

r20

∫ez · n dS = r20

∫ π /2

0cos θ sin θ dθ dϕ

= r202π[12sin2 θ

]π /20

= πr20

The term on the left-hand side of Equation 8.245 is a bound current source arising fromthe polarisation of the sphere due to the external applied electric field. The first term onthe right is the bound current passing through the sphere surface and the second term isthe external ‘dipole current’ looping back to the source (Figure 8.22). The three termsof Equation 8.245 express the Kirchhoff current law (Section 8.2.1): that all currentsentering and leaving a node in a circuit must sum to zero.

8.9.1.1 Dielectric sphere

We suppose for the moment that εin represents an (almost) lossless dielectric. In thatcase the permittivity of the sphere is complex but with a negligible imaginary termand εin > ε0. The circuit of the displacement current can then be represented as in


Rsphere

Csphere Cdipole

Isphere

Isource Idipole

Figure 8.23 Equivalent circuit for a sub-wavelength dielectric sphere subject to an appliedoscillating electric field. The displacement cur-rents into and out of the round node at thebranch obey Kirchhoff’s current rule. The di-electric sphere is represented by a capacitor inparallel with a very low-loss resistor, and the di-pole loop displacement current is also representedby a capacitor.

Figure 8.23. Now from the electric field at the surface of the sphere we can find thepotential difference across the sphere. The potential is given by

Vsphere = –∫σ

Esurf · dl

= –2∫ 0

–π /2


]E0 cos θ r0 sin θ dθ

= r0


]E0 (8.246)

Now from Vsphere and Isphere we can identify the characteristic impedance of the sphere as

Zsphere =Vsphere

Isphere

=i

ωεinπr0(8.247)

and the impedance of the dipolar field is

Zdipole =Vsphere

Idipole

=i

ωεout2πr0(8.248)

From the phasor form of the impedance (Section E.2 in Appendix E) we have

Z = R + i (XL +XC) (8.249)

XL = –ωL inductive reactance (8.250)

XC =1ωC

capacitive reactance (8.251)


Comparing Equation 8.251 with Equations 8.247 and 8.248 we see that

Csphere = εinπr0 (8.252)

Cdipole = εout2πr0 (8.253)

8.9.1.2 Metallic sphere

In the case of a metallic sphere, the real part of the permittivity can be strongly negativeand the resistive loss non-negligible. In that case, the equivalent circuit can be construedas shown in Figure 8.24. The sphere impedance becomes

Zmetalsphere =

iωRe[εin]πr0

(8.254)

and comparing Equation 8.254 with Equation 8.250 we see that

Lmetalsphere = –

iω2Re[εin]πr0

(8.255)

Thus, we see that the nanosphere can act as a capacitive element or an inductive elementby choosing the material property: insulating or conducting.

8.9.2 Equivalent circuit for plasmon surface waves

In this section we reconsider surface waves at the interface between a dielectric andmetal. We apply what we have learned about planar transmission lines, their relation toplane waves (Section 8.3.4), and the equivalent circuits that can represent a transmissionline. The basic idea is to treat a plane wave incident normally on the interface as a planartransmission line and the interface itself as a junction between two transmission lines:one in the dielectric and the other in the metal.

8.9.2.1 Transmission line equivalent circuit

A plane wave impinging on a dielectric-metal interface can be modelled as a real trans-mission line. Figure 8.25 shows in the left panel the plane wave incident normal to the

Isource Idipole

Cdipole

Rsphere

Lsphere

Isphere

Figure 8.24 Equivalent circuit of a subwavelengthmetallic sphere subject to an external oscillating elec-tric field. The source displacement current drivesthe sphere and Kirchhoff’s current rule applies tothe branch point node. Current through the sphereexhibits inductive reactance in parallel with a non-negligible resistive loss. The dipole branch exhibitscapacitive reactance as in the case of the dielectricsphere.


(a)

Plane wave

Dielectric

Metal

(b)

Dielectric

Metal

Zs

Zs

Zp

Figure 8.25 Comparison between plane wave and equivalent impedanceT-network. Panel (a) Plane wave incident on a dielectric-metal interface.Panel (b) Equivalent circuit for a transmission line representing the planewave.

interface and the right panel the equivalent impedance T-network representing a trans-mission line. The expressions for the impedances are (see Reference 4 in Section 8.12),

Zs = Zc tanh(βl2

)(8.256)

Zp =Zc

sinh(βl)(8.257)

where

Zc =γc

iωεc(8.258)

and

γ 2c = β2 – εck20 = –k2c (8.259)

where β is the complex propagation constant, l the length of the transmission line, Zcthe characteristic impedance, and εc the characteristic dielectric constant of the medium.Using these expressions for the series impedance Zs and parallel impedance Zp, and


applying the rules for finding the ABCD matrix elements of the T-circuit we developedin Section 8.8.5, we find that for long transmission lines (l >> 1/β):

A = coshβl (8.260)

B = Z0 sinhβl (8.261)

C =1Z0

sinhβl (8.262)

D = coshβl (8.263)

in accordance with the expressions listed in Table 8.4. Now as the length of the trans-mission line tends to infinity, Z → 0, and in the limit of very long lines we have thesituation shown in Figure 8.26. In fact, the plane wave propagates through the dielec-tric and the metal, although the propagation length in the metal penetrates only as faras the skin depth. Therefore, we can characterise the light incident on the interface astwo transmission lines connected at the interface with two characteristic impedances, Zd0and Zm0 , for the dielectric and metal, respectively. Figure 8.27 shows the correspondencebetween plane wave propagation in the dielectric and metal, and the equivalent circuitof the interface. Now the transmission line impedances can be written in a particularlyuseful form if we consider Maxwell’s equations in a Laplace-transformed space. TheLaplace transform is an integral operator of the form

L =∫ ∞0

e–γ z dz (8.264)

(a)

Dielectric

Metal

Dielectric

Metal

Dielectric

Metal

(c)(b)

Zs Zs

Zs

ZsZs

Zp

Figure 8.26 The limiting form of the equivalent circuit of a long transmission line.


(a)

Plane wave

Dielectric

Metal

Dielectric

Metal

x

(b)

Z0d

Z0m

z

Figure 8.27 Correspondence between plane waves propagating in dielectric andmetal, Panel (a), and two equivalent circuit transmission lines joined at thedielectric-metal interface, Panel (b).

and transformation of field A(x, z) is expressed by,

A(γ0) = LA(x, z) =∫ ∞0

A(x, z)e–γ z dz (8.265)

with

γ 2 = –k2z (8.266)

where γ may be complex to take into account transmission line losses and kz the propa-gation parameter in the z direction. Details of this transformation and its consequencesare discussed in Reference 6 in Section 8.12. Here, we simply state one of the key resultsfor the Maxwell–Ampère law in Laplace space for incident waves with TM polarisation:

γ Hy = iωεEx (8.267)

where Hy and Ex are the Laplace-transformed y and x components of the H- and E-field, respectively; ω and ε have their usual meanings of frequency and permittivity. We


identify the characteristic impedance of the transmission line with the correspondingimpedance of the Laplace-transformed wave:

Z0 =ExHy

=γ

iωε(8.268)

Any surface waves travelling in the x direction must have the same propagation param-eter β in the dielectric and metal materials. With k0, the wave propagation vector in freespace, we have from a k-vector addition on each side of the interface:

γd =√β2 – εdk20 = ikdz (8.269)

γm =√β2 – εmk20 = ikmz (8.270)

The picture we have now is of two transmission lines along z that join at the dielectric-metal interface. The characteristic impedances for the dielectric and metal transmissionlines are

Zd0 =γd

iωεd(8.271)

Zm0 =γm

iωεm(8.272)

8.9.2.2 Transmission line transverse resonance

When a transmission line is populated by a wave that is just at the cutoff condition,kc, a standing wave is present in the line, and no net energy propagates along the line.A transmission line in this state is said to be in ‘resonance’. The two transmission lines,along z in our problem, are transverse to the interface x. In order for stable surfacewaves to propagate along the interface, they must be subject to the same restriction thatno energy propagates along z. Therefore, a necessary condition for stable, propagatingsurface waves along x is that our transmission lines along z fulfil the resonance condition.Since no net wave propagates along z, the net propagation parameter along z mustbe null. Therefore, according to Equations 8.271 and 8.272, a necessary condition fortransverse resonance in our two joined transmission lines is

Zd0 = Zm0 for all z (8.273)

Therefore:

γd

iωεd=

γm

iωεm(8.274)


or

kdzεd

=kmzεm

(8.275)

This ‘impedance matching’ condition at the interface, arising from the transmissionline at resonance, is equivalent to the field-matching condition that we found in Sec-tion 7.3, Equation 7.32. Thus, we see that a transmission line point of view gives us asupplementary insight into the physical conditions required for stable surface waves.

8.9.3 Equivalent circuit of a dielectric slit in a metal layer

In this section we consider the equivalent circuit of a dielectric slit milled in a metallayer. We suppose that both the thickness of the metal layer and the slit width are sub-wavelength. Figure 8.28 shows a schematic of the physical slit and the correspondingimpedance equivalent circuit. The equivalent circuit consists of the T-circuit, which weconsidered earlier in Section 8.9.2 for the slit and two symmetrical voltage sources, andimpedances that represent voltage and current running in the metal near the surfaces.We posit that a plane wave propagating along z is incident on the top surface and setsup a standing wave there. We saw in Section 3.2.3 that the magnetic field component ofthe wave is adjacent to the surface and penetrates to the skin depth within a real metal.This time-harmonically oscillating magnetic field, via Faraday’s law, induces currentsand voltages within the skin depth on the top surface and on the slit walls. A detailed dis-cussion of the currents circulating in the skin depth near the slit is given in Reference 7.in Section 8.12.

The first step in the analysis is to assume that the circuit is symmetrical with the metalwall impedances Z1 = Z2 = Zm. Then we consider the circuit with only one source, saythe one on the left associated with Z1 in Figure 8.28, and effectively ‘short-circuit’ theother source on the right. The resulting circuit consists of a single voltage source and aseries-parallel network of impedances that can be combined into a single impedance ZT .The expression for ZT is

ZT =[(Zm + Zsd)Zpd + (Zm + Zsd)(Zm + Zsd + Zpd)

Zm + Zsd + Zpd

](8.276)

Now we impose the ‘resonant transmission line’ condition to ensure that no poweris propagated perpendicular to the slit walls (along the x direction). As we saw inSection 7.6, power is propagated only along z with exponential fall-off of the fieldspenetrating the metal walls of the slit. This condition is that ZT = 0, which results inthe following impedance expression:

(Zsd + Zpd

)2 + (2Zm)(Zsd + Zpd

)+ Z2

m – Zm + Z2pd = 0 (8.277)


Dielectric

W

z

x

Metal

MetalMetal

Metal

Z1 Z2Zpd

Zsd Zsd

Figure 8.28 Schematic of a subwavelength slit of width w milled into asubwavelength metal layer. The dielectric in the slit is the same as thedielectric above and below the metal surface. In the usual case the metal isthe same on both sides of the slit so Z1 = Z2. The equivalent circuits of thetwo dielectric-metal interfaces and the T-circuit of the slit itself are shownin the impedance equivalent circuit.

Adapting the definitions in Equations 8.256–8.259, we have

Zsd = Zd tanhγdW2

(8.278)

Zpd =Zd

sinh γdW(8.279)

where

Zd = ± γd

iωεd(8.280)

and

γ 2d = β2 – εdk20 = –k2d (8.281)

withW the width of the slit. Substituting these definitions into Equation 8.277 we arrive,after some algebra, at

tanh γdW = ± 2ZdZmZ2d + Z

2m

= ± 2 (Zm/Zd)

1 + (Zm/Zd)2(8.282)

This expression is the transcendental equation that specifies the allowed modes in the slitthat propagate in z and are stable against radiation along x. Now recalling the identity,

Exercises 209

tanh x =2 tanh (x/2)

1 + [tanh (x/2)]2(8.283)

and comparing with Equation 8.282 we can identify that

tanhγdW2

= ±ZmZd

(8.284)

From the definition of the hyperbolic tangent in terms of exponential functions, this lastexpression can be further simplified to

eγdW[Zd + ZmZd – Zm

]= ±1 (8.285)

This is the final expression for the slit transcendental function in terms of the equivalentcircuit quantities. It is consistent with the transcendental equations found for the slit inChapter 7, obtained from a wave treatment.

8.10 Summary

The chapter begins with a review of Kirchhoff ’s rules for voltage and current that formthe basis of conventional circuit theory. Lumped circuit analysis of transmission linesis next followed by the prototypical lossless, plane-parallel transmission line. The closecorrespondence between plane waves and transmission lines is emphasised. Special ter-mination cases and line impedances close the discussion on transmission lines, which isreally a precursor to waveguides. Waveguides are divided into two types: TE modes andTMmodes (not to be confused with TE and TM polarisation). Propagating solutions tothe two common geometries, flat slab and cylindrical waveguides, are worked out in de-tail. The next section treats networks of waveguides. The discussion then shifts to nano-structures and equivalent circuits, including plasmonic waveguiding in a slit geometry.

8.11 Exercises

1. Suppose two lossless transmission lines with different characteristic impedances arecoupled. If Z1 = 50� and Z2 = 75�, calculate the fractional power reflected andtransmitted at the interface between the two lines.

2. For a transmission line of length λ/4, calculate the impedance of the line if it is:(a) shorted, (b) open-circuited.

3. For a rectangular waveguide, calculate the cutoff wave vector kc for TM modes if theguide is square with side s = 800nm and if the guide is very thin with s1 = 800 nm ands2 = 100 nm. Assume the guide is excited by a plane wave of λ = 500nm impingingnormally on the front face of the guide. Calculate the characteristic impedance of thistransmission line.


4. Consider the interface between silver and glass. Assume that 1mW power is coupledto the interface and converted to surface plasmon polariton (SPP) waves. Fromwhat you learned in Chapter 7, calculate the stable surface wave propagation par-ameter assuming the exciting field wavelength is 832 nm. Use the following dielectricconstants: glass ε′ = 2.40, silver ε = –32.8 + i0.46.

5. Using the same data as in Exercise 4., calculate the impedance Zp of the equivalentT-circuit and the characteristic impedance Z0 of the corresponding transmission line.

6. Calculate the TE and TM modes that can propagate in a planar metal-dielectric-metal (MDM) waveguide with water between the two metal walls. The distancebetween the two metal slabs is 0.25 μm and the free-space wavelength is 1.55μm.The dielectric constant of water at λ0 = 1.55μm can be considered real and equal to1.77.

7. Calculate (a) the capacitance of a glass nanosphere with radius equal to 50 nm and(b) the capacitive reactance of the sphere at an excitation wavelength of 832 nm. Thedielectric constant of glass is 2.25.

8. Suppose we have a sphere with the same dimensions as in Exercise 7 but fabricatedfrom silver metal. Using the dielectric constant data of Exercise 4, calculate (a) theinductance of the sphere and (b) the inductive reactance.


1. D. M. Pozar, Microwave Engineering, 3rd edition, John Wiley & Sons, Hoboken, NJ(2005).

2. S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in CommunicationElectronics, 3rd edition, John Wiley & Sons, New York (1994).

3. N. Engheta, A. Saladrino, and A. Alù, Circuit elements at optical frequencies: nano-inductors, nano-capacitors, and nano-resistors. Phys Rev Lett vol 95, p. 095504 (2005).

4. C. D. Papageorgiou and J. D. Kanellopoulos, Equivalent circuits in Fourier space forthe study of electromagnetic field. J Phys A: Math Gen vol 15, pp. 2569–2580 (1982).

5. S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, Transmission Line and Equiva-lent Circuit Models for Plasmonic Waveguide Components. IEEE J Sel Topics Quant Elecvol 14, pp. 1462–1472 (2008).

6. F. Nunes and J. Weiner, Equivalent Circuits and Nanoplasmonics. IEEE T. Nano vol 8,pp. 298–302 (2009).

7. J. Weiner, The electromagnetics of light transmission through subwavelength slits inmetallic films. Opt Express vol 19, pp. 16139–16153 (2011).

9

Metamaterials

9.1 Introduction

The term ‘metamaterials’ describes objects composed of conventional materials but withat least one length dimension well below an optical wavelength. The subwavelength scalein the structural composite gives rise to optical properties unique to the structure anddistinct from the bulk properties of the individual components. This topic is an active,rapidly developing research area, and therefore, this chapter must be considered morea status report than an exposition of canonical received wisdom. We discuss the distin-guishing features of light interacting with metamaterials with emphasis on their simplestrealisations: slab waveguides and periodic layered material in which strong permittiv-ity modulation from positive (dielectrics) to negative (metals) gives rise to uncommonoptical behaviour. We begin with a brief discussion of two of these unconventional phe-nomena that have figured importantly in the motivation for metamaterials development:left-handed materials and negative refractive index [1]. The propagation of light throughstacked layers is then treated first at conventional length scales [2] and then again in thesubwavelength domain.

9.2 Left-handed materials

An elementary expression in physical optics relates the phase velocity of an electromag-netic wave propagating through a material to the permeability μ and permittivity ε ofthat material:

v =1√με

(9.1)

When light propagates in vacuum the relation is

c =1√μ0ε0

(9.2)


212 Metamaterials

Normally one only considers the positive solution of the square root, but in fact if μand ε are both negative, then their product is still positive but the negative root can (andshould) be assigned to the phase velocity. This possibility was discussed early on in aseminal paper by Veselago [3]. For plane harmonic waves in an isotropic material thewave vector k is related to v through the frequency ω by

k =ω

v(9.3)

and consequently if v < 0 then k < 0. Since k = nk0, where n is the refractive index of thematerial and k0 = ω/c, the refractive index of a material with negative permeability andpermittivity is also negative. Suppose we have a plane wave propagating in an isotropicmaterial such that the E,H fields are oriented along the x, y axes in the positive direction:

Ex = E0eikzze–iωt (9.4)

Hy = H0eikzze–iωt (9.5)

Then according to Maxwell’s curl equations

kzEx = ωμHy (9.6)

kzHy = ωεEx (9.7)

Clearly if μ, ε > 0 we have a conventional ‘right-handed’ relation between Ex,Hy, kz. Ifμ, ε are both negative, however, kz points along –z when Ex,Hy point along x, y; and therelation among the three vectors is said to be ‘left-handed’. Figure 9.1 shows the relation-ship between E,H , k in right-handed and left-handed systems. A ‘left-handed’ materialis therefore one with both μ, ε < 0. In another seminal article, Pendry [4] pointed outthat a slab of left-handed material would focus propagating and evanescent waves eman-ating from a source, and therefore would be the realisation of a ‘perfect’ lens, unfetteredby the usual imaging diffraction limit.

Up to this point analysis of the properties of left-handedness in optics was entirelyspeculative because no known natural material possessed the necessary property ofboth negative permeability and permittivity. However, if the spatial gradients of theE,H fields are very small (i.e. the characteristic spatial dimensions are ‘subwavelength’)

x

y

z

Ex

Hykz

x

y

z

Ex

Hy kz

(a) (b)

Figure 9.1 (a) Right-handed material.(b) Left-handed material.

Negative index metamaterials and waveguides 213

then we get an important simplification in the equations governing propagation. Weremember from Equations 7.3–7.8 of Chapter 7 that

∂Ey∂z

= –(iωμ0μ)Hx

∂Hx

∂z–∂Hz

∂x= –i(ωε0ε)Ey

∂Ey∂x

= (iωμ0μ)Hz

TE polarisation

and

∂Hy

∂z= (iωε0ε)Ex

∂Ex∂z

–∂Ez∂x

= (iωμ0μ)Hy

∂Hy

∂x= –(iωε0ε)Ez

TM polarisation

In the subwavelength regime the second equation in each set of TE and TM po-larisation will be much smaller than the other two and can be dropped. We see,therefore, that waves TE polarised depend only on the permeability and waves TMpolarised depend only on the permittivity. Negative permittivity does exist in availablematerials in the optical frequency range, notably good conductors such as gold andsilver. Therefore, in order to demonstrate a negative refractive index we only need tostudy metamaterial subwavelength structures with overall negative permittivity in somefrequency range, even if the permeability remains essentially equal to μ0.

9.3 Negative index metamaterials and waveguides

The simplest structures that fulfil these requirements are two-dimensional waveguidesconsisting of a dielectric core and metal cladding, termed metal-insulator-metal (MIM)waveguides, or the opposite arrangement of a metal core and dielectric cladding, (IMI)waveguides. A schematic diagram of a typical MIM waveguide is shown in Figure 9.2.The waveguide consists of a high-index, low-loss core, in this case gallium phosphide(GaP), sandwiched between two silver (Ag) claddings. A plane-wave or modal-wavelight source emits in the z direction and excites waveguide modes propagating along z.These waveguide modes are actually surface plasmons excited at the two metal-dielectricinterfaces and coupled to form symmetric and antisymmetric waveguide modes with re-spect to the z centreline. The optical properties of these waveguides have been studied ina series of articles [5–8], and Figure 9.3 summarises the dispersion relations for Ag-GaP-AgMIM structures for three different core thicknesses. Ignoring for the moment the factthat these modes are lossy when implemented with real metals, we can determine the re-gions of negative index behaviour from inspecting the real dispersion curve, Figure 9.3a.A well-known result from physical optics identifies the phase velocity of a harmonic wave

214 Metamaterials

Ag AgGaP

t

source

z

x

Figure 9.2 Diagram of a typical MIM waveguide.The core thickness t is subwavelength (usually some tensof nanometres) and the metal cladding width is a factorof ten or more greater than the core. The source can bea plane wave or an amplitude modulated ‘modal’ wavewith appropriate symmetry to excite a desired waveguidemode, symmetric or antisymmetric.

X

0 0.35Imag (kz)nm–1

0 0.5

25 nm17 nm

10 nm

25 nm17 nm

10 nm–0.5

1.5

2

2.5

3

Real (kz)nm–1

Ene

rgy

(eV

)

X

HyZ

(a) (b)

(c)

Figure 9.3 (a) Dispersion diagram (Re(kx) vs. mode energy) for waveguide modes supportedby waveguide structure shown in Figure 9.2. (b) Same as (a) for Im(kx). Blue curves correspondto symmetric modes; red curves to antisymmetric modes with three different widths, t. Thehorizontal dotted line is the energy of the surface plasmon resonance. (c) Waveforms for the twolowest-energy antisymmetric and symmetric modes. Figure adapted from Figure 2 of [8] andused with permission.

travelling in the z direction as vϕ = ω/kz and the group velocity (equivalent to the energyvelocity for lossless, dispersionless media) as vγ = dω/dkz. When the phase velocity andgroup velocity have opposite signs, a lossless medium is said to exhibit negative indexbehaviour. Thus, we see from Figure 9.3a that the lower right quadrant of the diagramindicates positive phase velocity and positive group velocity for the symmetric modes;while the upper left quadrant shows negative phase velocity and positive group velocityfor antisymmetric modes. The antisymmetric modes in this quadrant exhibit negativeindex behaviour.

Negative index metamaterials and waveguides 215

In the more general case of lossy media, the propagation vectors become complexand the association of group velocity with energy or flux velocity becomes ambiguous.However, we can always associate the energy flux of any electromagnetic wave with thePoynting vector S = E ×H. Therefore, the more general criterion, including lossy ma-terials, is that if the real part of S and vϕ propagate in opposite directions, the waveguidemode exhibits negative index behaviour. In order for the modes to have any practicalsignificance, the imaginary part of kz must be small compared to the real part. Assum-ing that propagation along z is represented by a factor eikzz it is clear that Im(kz > 0,Re(kz < 0, and Im(kz)/Re(kz) 1 are equivalent criteria for significant negative in-dex modes. Inspection of Figure 9.3a and b shows that these criteria are satisfied forasymmetric modes above the plasmon resonance energy (upper left-hand quadrant,Figure 9.3a), consistent with our finding for the equivalent lossless case.

Hy Amplitude Antisymmetric Mode in XZ Plane

X-axis (nm)

Z-a

xis

(nm

)

–50 –40 –30 –20 –10 0 10 20 30 40 50

50

100

150

200

250

300

350

400

450

500

Ag Ag GaP

Figure 9.4 Real part of Hy amplitude of the lowest energy antisymmetric waveguide mode in theX – Z plane for a Ag-GaP-Ag structure with a core width of 25 nm. Red indicates positive amplitudeand blue indicates negative amplitude. The figure illustrates the Hy amplitude antisymmetry along zwith respect to the node at x = 0.

216 Metamaterials

–2.5 –2 –1.5 –1 –0.5 0 0.5 1 1.5 2 2.5× 10–7

–300

–200

–100

0

100

200

300

400

Z-axis (m)

Re(

Hy)

Re(Hy) vs Z Ag-GaP-Ag MIM

Figure 9.5 Plot of Re(Hy) as a function of z along the waveguide at x = 12.5 nm. The decrease inamplitude along z is due to dissipation from the lossy real metal.

Figures 9.4 and 9.5 show the results of an FDTD (Finite Difference Time Domain)numerical simulation of the lowest-energy antisymmetric mode propagating in a Ag-GaP-Ag waveguide with a core width of 25 nm. The real part of the amplitude of Hy intheX–Z plane is shown in Figure 9.4 and the propagation along z is shown in Figure 9.5.

9.4 Reflection and transmission in stacked layers

9.4.1 Matrix formulation

In Section 8.8.5 of Chapter 8 we discussed a matrix approach to relate input and outputvoltages and currents in linear circuits. A similar approach can be used to relate trans-mission and reflection, first treated at a single interface in Chapter 3 Section 3.2, acrossan optical ‘network’ consisting of material layers with differing indices of refraction.

Reflection and transmission in stacked layers 217

X

Z = d

Z = 0

ε1

A1

A2

A3

A2

B1

B2

B3

B2

ε2

ε3

Z

Figure 9.6 Diagram of the three-layer stratified material withpermittivities ε1, ε2, ε3. Wavy arrows indicate plane waves propa-gating, reflecting, and transmitting at the z = 0 and z = d interfaces.The A,B coefficients indicate the wave amplitudes at the interfaces.

Figure 9.6 shows a plane wave entering a three-layer stratified medium from the regionz < 0 with permittivity ε1, partially transmitting and reflecting at the z = 0 boundary,propagating from z = 0 to z = d in the layer with permittivity ε2, and then once againundergoing reflection and transmission at the z = d boundary between layer 2 and layer3 with ε3. Consider the electric field of the wave propagating in the XZ plane:

E = E(z)ei(nk0x–ωt) (9.8)

with n the index of refraction and n =√ε. The properties of the layers are invariant

along the axes orthogonal to z, and we assume the wave polarised TM (Hy,Ex,Ez) orTE (Ey,Hx,Hz). In the following discussion we will concentrate on TM polarisationbecause it gives rise to surface plasmons that play an important role in many import-ant phenomena involving metamaterials. Expressions for TE polarisation are left to thereader as an exercise. The propagation angles k · r = kxx + kzz are not necessarily nor-mal to the boundaries and may be reflected and transmitted at angles according to theFresnel laws (see Section 3.2). The components kz and kx are related to k:

k2x + k2z = k2 (9.9)

In the electrical and optical engineering literature [9] the component of the wave vectortransverse to the propagation direction is often denoted by β:

kx = β =√k2 – k2z (9.10)

Dropping the time dependence for the moment, we can write the travelling wave movingup and down along the z direction,

218 Metamaterials

E(z) = Ue(ikzz) +De–(ikzz) = A(z) + B(z) (9.11)

where A,B are the complex amplitudes of the waves. On each side of the z = 0 andz = d boundaries there are two amplitude pairs. Specifically at the z = 0 boundarywe have A1,A′2 and B1,B′2. These two amplitude pairs must be related by the Fresnellaws of reflection and transmission and can be expressed as two-element column vectorstransformed by a 2× 2 matrix D12:(

A1

B1

)= D12

(A′2B′2

)(9.12)

where D12 itself consists of a product of two other matrices, the inverse of a matrix D1

and another matrix D2. Thus:

D12 = D–11 D2 (9.13)

where D1,D2 express the field continuity conditions at the z = 0 boundary. Figure 9.7shows in detail, reflection and transmission at the z = 0 boundary from a plane waveincident at angle θ1 from z < 0. Referring to Figure 9.7 and restricting the discussion toTM polarisation we write the continuity conditions for the Ex and Hy field components:

–E1 cos θ1 + E ′1 cos θ1 = –E2 cos θ2 + E ′2 cos θ2 (9.14)√ε1E1 +

√ε1E′1 =

√ε2E2 +

√ε2E′2 (9.15)

where we have used Hy =√ε/μEx and have assumed non-magnetic material with μ = 1.

In matrix form the continuity relations are

A2θ2

θ1

θr

E1

B2

E2

n2

n1

E1

X

A1 B1

–H1 –H1

–H2

E2

–H2

Z

Figure 9.7 Detail of the incident, reflected, and transmit-ted field components for TM polarisation at the z = 0. Notethe phase convention that Ex changes sign on reflection. Notealso that the Hy components are aligned so that the Poyntingvectors point in the directions of propagation.


D1

(E1

E ′1

)= D2

(E2

E′2

)(9.16)

(– cos θ1 cos θ1√

ε1√ε1

)(E1

E ′1

)=(– cos θ2 cos θ2√

ε2√ε2

)(E2

E′2

)(9.17)

Multiplying both sides of Equation 9.16 by D–11 results in Equation 9.13. Writing out the

D12 matrix explicitly, we have,

(E1

E ′1

)= D–1

1 D2 = D12 =12

⎡⎣ cos θ2

cos θ1+√ε2ε1

– cos θ2cos θ1

+√ε2ε1

– cos θ2cos θ1

+√ε2ε1

cos θ2cos θ1

+√ε2ε1

⎤⎦(E2

E′2

)(9.18)

The incident and transmission angles cos θ1, cos θ2 can be expressed in terms of thepropagation vector components kz1, kz2, the refractive indices n1, n2, or permittivitiesε1, ε2 on each side of the boundary:

kz1 = n1k0 cos θ1 and kz2 = n2k0 cos θ2 (9.19)

and

cos θ2cos θ1

=n1kz2n2kz1

=√ε1

ε2

kz2kz1

(9.20)

Then D12 can be expressed as

D12 =12

√ε2

ε1

[ kz2kz1

ε1ε2

+ 1 – kz2kz1ε1ε2

+ 1

– kz2kz1ε1ε2

+ 1 kz2kz1

ε1ε2

+ 1

](9.21)

and evidently

D23 =12

√ε3

ε2

[ kz3kz2

ε2ε3

+ 1 – kz3kz2ε2ε3

+ 1

– kz3kz2ε2ε3

+ 1 kz3kz2

ε2ε3

+ 1

](9.22)

Again referring to Figure 9.7, and after setting E ′2 = 0, we can express the Fresnelreflection and transmission coefficients as

r12 =– kz2kz1

ε1ε2

+ 1kz2kz1

ε1ε2

+ 1and t12 =

2√ε2ε1

(kz2kz1

ε1ε2

+ 1)

r23 =– kz3kz2

ε2ε3

+ 1kz3kz2

ε2ε3

+ 1and t23 =

2√ε3ε2

(kz3kz2

ε2ε3

+ 1) (9.23)

220 Metamaterials

From which we can write D12 as

D12 =1t12

[1 r12r12 1

](9.24)

and similarly

D23 =1t23

[1 r23r23 1

](9.25)

In the region 0 < z < d the plane wave propagates freely with no change in amplitudeor direction. From Figure 9.6 we can write the propagation matrix P as(

A2

B2

)= P2

(A′2B′2

)=(eiϕ 00 e–iϕ

)(A′2B′2

)(9.26)

The propagation phase ϕ = kz2d, where d is the thickness (along z) of the second layer.Similarly to the boundary at z = 0, we have at the boundary z = d:(

A2

B2

)= D–1

2 D3

(A′3B′3

)= D23

(A′3B′3

)(9.27)

We see from Equations 9.12,9.26, and 9.27 that the amplitudes just above z = d can berelated to those just below z = 0 by a sequence of matrix multiplications:(

A1

B1

)= D12P–1

2 D23

(A′3B′3

)= D–1

1 D2P–12 D

–22 D3

(A′3B′3

)(9.28)

This sequential scheme can easily be generalised to a multilayer stack starting withamplitudes Ai ,Bi and ending with amplitudes A′f ,B

′f :

(AiBi

)= D–1

i

⎡⎣ N∏j=1

D–1j P

–1j Dj

⎤⎦Df

(A′fB′f

)(9.29)

and we define the matrixM as

M = D–1i

⎡⎣ N∏j=1

D–1j P

–1j Dj

⎤⎦Df (9.30)

and (AiBi

)=(M11 M12

M21 M22

)(A′fB′f

)(9.31)


Taking into account that after the light exits the final interface, B′f = 0, we can expressthe overall reflection and transmission coefficients r, t as

r =BiAi

(9.32)

t =A′fAi

(9.33)

and using Equation 9.31:

r =M21

M11and t =

1M11

(9.34)

In the relatively simple case of a three-layer stack we can write out explicitly the elem-ents of the M matrix. The matrix multiplications are most easily carried out using theexpressions for D12 and D23 in terms of the reflection and transmission coefficientsr12, r23, t12, t23, in Equations 9.24 and 9.25:

M =1

t12t23

[e–iϕ2 + r12r23eiϕ2 r23e–iϕ2 + r12eiϕ2

r12e–iϕ2 + r23eiϕ2 r12r23e–iϕ2 + eiϕ2

](9.35)

Then using Equation 9.34:

r =r12e–iϕ2 + r23eiϕ2

e–iϕ2 + r12r23eiϕ2(9.36)

t =t12t23

e–iϕ2 + r12r23eiϕ2(9.37)

Substituting for r12, t12 and r23, t23 into Equation 9.23:

r =

(1 – kz3

kz1ε1ε3

)cosϕ2 +

(kz2kz1

ε1ε2

– kz3kz2

ε2ε3

)i sinϕ2(

kz3kz1

ε1ε3

+ 1)cosϕ2 –

(kz2kz1

ε1ε2

+ kz3kz2

ε2ε3

)i sinϕ2

(9.38)

and

t =2√

ε3ε1

[(kz3kz1

ε1ε3

+ 1)cosϕ2 –

(kz2kz1

ε1ε2

+ kz3kz2

ε2ε3

)i sinϕ2

] (9.39)

222 Metamaterials

If the three-layer stack is symmetric with ε1 = ε3 then the expressions for r and t simplifysomewhat

r =

(kz2kz1

ε1ε2

– kz1kz2

ε2ε1

)i sinϕ2

2 cosϕ2 –(kz2kz1

ε1ε2

+ kz1kz2

ε2ε1

)i sinϕ2

(9.40)

and

t =2

2 cosϕ2 –(kz2kz1

ε1ε2

+ kz1kz2

ε2ε1

)i sinϕ2

(9.41)

9.4.2 Periodic stacked layers

We consider here the case of alternating periodic layers of two permittivities. The ana-lysis of periodic stacked layers has much in common with condensed matter theory ofcrystalline materials. The periodicity and Fresnel scattering give rise to transmission win-dows, stopbands, and ‘photonic band gaps’ analogous to the physics of metals and dopedsemiconductors [10],[11]. The development builds on the matrix approach introducedin Section 9.4.1. Figure 9.8 illustrates the situation. The permittivity varies periodicallyalong z:

ε(z) = ε1 0 < z < d

ε(z) = ε2 d < z < � (9.42)

ε2

ε1

ε2

ε1

ε2

ε1

ε2

ε1

(n+2)Λ

(n+1)Λ

(n–1)ΛAn–1

An+1

An+2

Cn+1

Cn+2

Cn+3

Cn

X

Z

An

Bn–1

Bn+1

Bn+2

Dn+1

Dn+2

Dn+3

Dn

Bn

(n)Λ

d

Λ-d

Figure 9.8 Periodic stacked layers of two different permittiv-ities, ε1, ε2. The repeating unit has a length of �. Matricescorresponding to material 1 are denoted A,B, and matricescorresponding to material 2 are denoted C,D.


where � is the period of the alternating layers. We can write the E-field travelling wavealong the stack similarly to Equation 9.11:

E(z) = Aneikz1[(z–n�)] + Bne–ikz1[(z–n�)] n� – d < z < n� (9.43)

E(z) = Cneikz2[(z–(n�–d)] +Dne–ikz2[(z–(n�–d)] (n – 1)� < z < n� – d (9.44)

where k1, k2 are related in the usual way to their longitudinal and transverse componentspropagating in material slabs with permittivities ε1, ε2:

kz1 =√k21 – β

2 = k1 cos θ1

kz2 =√k22 – β

2 = k2 cos θ2 (9.45)

The β term is the transverse component defined in Equation 9.10.Just as in Equations 9.12, 9.13 9.16, and 9.26, the matrix form the E-field coefficients

are related by

(An–1Bn–1

)= D–1

1 D2P–12

(Cn

Dn

)(9.46)

and (Cn

Dn

)= D–1

2 D1P–11

(AnBn

)(9.47)

where the propagation matrices within the slabs ε1 and ε2 are written as

P1 =(eikz1d 00 k–ikz1d

)(9.48)

and

P2 =(eikz2(�–d) 0

0 k–ikz2(�–d)

)(9.49)

Substituting Equation 9.47 into Equation 9.46, we obtain a matrix that relates theincident and reflected amplitudes from one unit cell to the next:

(An–1Bn–1

)= D–1

1 D2P–12 D

–12 D1P–1

1

(AnBn

)(9.50)

(An–1Bn–1

)=[A BC D

](AnBn

)(9.51)

224 Metamaterials

Carrying out the explicit matrix multiplications for TM polarisation, we have for theelements of the ABCD matrix,

A = e–iϕ2[cosϕ1 –

12i sinϕ1

(kz2kz1

ε1

ε2+kz1kz2

ε2

ε1

)](9.52)

B = eiϕ2[–12

(kz2kz1

ε1

ε2–kz1kz2

ε2

ε1

)i sinϕ1

](9.53)

C = e–iϕ2[12

(kz2kz1

ε1

ε2–kz1kz2

ε2

ε1

)i sinϕ1

](9.54)

D = eiϕ2[cosϕ1 +

12i sinϕ1

(kz2kz1

ε1

ε2+kz1kz2

ε2

ε1

)](9.55)

where ϕ1 = kz1d and ϕ2 = kz2(� – d).The reflection and transmission of light through periodic stacked layers is our pri-

mary interest, and therefore we develop expressions for Bragg reflection and ‘resonanttunnelling’ transmission. The latter is particularly important when the stack consists ofalternating layers of metal and dielectric because plasmon surface waves form at themetal-dielectric interfaces. These surface waves propagate along X ,Y and are evanes-cent along Z. We will see that, despite this evanescence, under the right conditions lightcan be transmitted through a periodic stack by resonant tunnelling.

Rewriting Equation 9.51 as a relation between the zeroth layer of an ε1 slab and thenext highest ε1 layer,

(A1

B1

)=[A BC D

]–1 (A0

B0

)(9.56)

we can interpret the inverse of the ABCD matrix as the operator that transfers the waveproperties from layer 0 to layer 1. In fact, once A0,B0 are specified, the inverse of theABCD matrix can transfer these properties to the ‘nth’ layer of ε1:(

AnBn

)=[A BC D

]–n (A0

B0

)(9.57)

The inverse of the ABCD matrix is given by

(A BC D

)–1

=1

det(ABCD)

(D –B–C A

)(9.58)

and it can be verified by direct substitution from Equations 9.52–9.55 that

AD – BC = 1 (9.59)


Therefore once the properties of the zeroth layer are fixed, they can be transferred to the‘nth’ layer by successive matrix multiplications of the inverse ABCD matrix:(

AnBn

)=[D –B–C A

]n (A0

B0

)(9.60)

The ABCD matrix (and its inverse) also sometimes called the ‘characteristic’ matrix orthe ‘transfer’ matrix, and the theory of field displacement through stacked or stratifiedlayers is often referred to as ‘transfer matrix theory’ or TMT[12], [13].

9.4.3 Bloch waves

A harmonic plane wave, E = E0ei(kz–ωt), travelling in vacuum or any uniform, isotropic,lossless medium, is periodic in time and space according to the wave angular frequency ωand speed of propagation, v. The simple dispersion relation k = ω/v specifies the spatialperiodicity. If the medium is the vacuum, k = ω/c. A wave travelling in a periodicmediummust exhibit the symmetry this added periodicity provides as well. The symmetry istranslational such that

EK (z) = EK(z +�) (9.61)

Such a wave is called a Bloch wave [14] and has the form

EK(z) = EKeiKz (9.62)

where K = 2π /� or, taking into account the propagation in the transverse direction x aswell,

EK(x, z) = EKeiβxeiKz (9.63)

In column vector form we have (AnBn

)=(eiK�

)n (A0

B0

)(9.64)

and comparing Equation 9.57 to Equation 9.64:[A BC D

]–n (A0

B0

)=(eiK�

)n (A0

B0

)(9.65)

we see that eiK� is an eigenvalue of the inverse ABCD translation operator matrix. For aone-step (unit periodic cell) ABCD translation in the –z direction:[

A BC D

](A0

B0

)= e–iK�

(A0

B0

)(9.66)

226 Metamaterials

For a one-step translation in the +z direction,

[A BC D

]–1 (A0

B0

)= eiK�

(A0

B0

)(9.67)

The eigenvalues of the matrix in Equation 9.66 can be written in terms of the matrixelements by solving the usual determinant equation:

det[A – λ BC D – λ

]= 0 (9.68)

so

e–iK� =12(A +D)±

√[12(A +D)

]2– 1 (9.69)

where we have used AD–BC = 1. In order for the Bloch wave on the left to be propagat-ing, the factor K� in the argument must be real. The wave consists of a real part cosK�and an imaginary part – sinK�. The expression on the right must be also be complex tosatisfy the equation, which implies that (A + D)/2 must be less than unity for the Blochwave to propagate through the periodic medium. If (A + D)/2 > 1, then the right-handside is real and the Bloch vector K must then be pure imaginary so that the exponentialargument becomes real and the wave evanescent. The sign of K is chosen to ensure thisevanescent behaviour. We can use Equations 9.52 and 9.55 to write out A + D in termsof the plane waves of which the Bloch wave is composed. For periodic structures we setd = d1,� – d1 = d2:

A+D = 2 cos(kz1d1) cos(kz2d2)–(kz2kz1

ε1

ε2+kz1kz2

ε2

ε1

)sin(kz1d1) sin(kz2d2) d1 +d2 = �

(9.70)For a fixed stack geometry (d1, d2) and choice of materials (ε1, ε2), as ω increases, therewill be zones of propagation and zones of evanescence. The non-propagating zones arethe ‘stopbands’ or ‘photonic band gaps’ and the propagating zones are analogous to theconduction bands in condensed matter theory. They are sometimes termed ‘photonicband windows’. The photonic band gaps are truly non-propagating only for periodicstructures extending infinitely in±z. For practical, finite structures the evanescent wavescan ultimately tunnel through and emerge propagating at the exit boundary. We shall seein Section 9.4.7 that resonant tunnelling in metamaterials is very important.

We write Equation 9.69 in the following form:

cosK� = cos(kz1d1) cos(kz2d2) –12

(kz2kz1

ε1

ε2+kz1kz2

ε2

ε1

)sin(kz1d1) sin(kz2d2) (9.71)


or

K =1�

cos–1[cos(kz1d1) cos(kz2d2) –

12

(kz2kz1

ε1

ε2+kz1kz2

ε2

ε1

)sin(kz1d1) sin(kz2d2)

](9.72)

where we remember that k1 = k0n1 = (ω/c)√ε1 and k2 = k0n2 = (ω/c)

√ε2 are the two

wave vectors resulting from a single-frequency wave k0 = ω/c incident on the periodicstructure. Equation 9.72 is the Bloch wave dispersion relation K(ω). The cosK� termon the left-hand side of Equation 9.71 may contain real or complex K . A cos functionwith imaginary argument becomes a hyperbolic cosine (cosh) function. In the propagat-ing region at the band gap edge, K� = π , and at the centre frequency of the band gapitself

K� = π ± ix (9.73)

where x is yet to be determined. Figure 9.9 plots the plane-wave frequency ω vs K� forthe case where the optical thickness of the two layers is equal (n1d1 = n2d2). Now weassume that, given our choice of geometry d1, d2 and materials ε1, ε2, a frequency ω0 canbe found such that the plane wave phase accumulation in the two slabs is

k1zd1 = k2zd2 =π

2(9.74)

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

KΛ (π)

ω (π

)

Figure 9.9 Plot of K� (π units) as a functionof a ‘normalised’ plane-wave frequency ωN =(n1d1/c)ω = (n2d2/c)ω (π units). The refractiveindices are n1 = 1.5 and n2 = 2.5 for slabs 1 and2. The optical thicknesses of the two slabs havebeen chosen to be equal (n1d1 = n2d2). The solid(dashed) trace is for the Bloch wave propagatingin the +z (–z) direction. The shaded area showsthe band gap where the Bloch wave is evanescent.

228 Metamaterials

To keep the development as simple as possible, we only consider normal incidence sothat kz1, kz2 = k1, k2 = n1k0, n2k0. Then, substituting Equation 9.73 into Equation 9.71:

cosh x =12

(n1n2

+n2n1

)(9.75)

Taking the inverse hyperbolic cosine of both sides, we find that

x � lnn2n1� 2 · n2 – n1

n2 + n2(9.76)

where we have continued to assume, for the time being, that ε1, ε2 represent losslessdielectrics and that the periodic permittivity modulation in the structure is not too great.

According to Equation 9.73, x is the imaginary part of K� in the centre of the bandgap. At the edge of the band gap, at the boundary between the propagating and non-propagating zones, x = 0. Let ω – ω0 be the deviation from the plane-wave frequencycorresponding to the centre of the band gap and η be the plane-wave phase anglecorresponding to this frequency deviation. We then write

η =n1d1cω –

π

2=n2d2cω –

π

2(9.77)

or

η +π

2=n1d1cω =

n2d2cω (9.78)

Now substitute Equations 9.78, 9.75, and 9.73 into Equation 9.71. The result is

cosh[x(ω)] =12

(n2n1

+n1n2

)cos2 η – sin2 η (9.79)

which is a relation expressing x as a function of ω. The plane-wave frequency at the bandedge ωbe corresponds to x = 0 and is given by

sin ηbe = ± n2 – n1n2 + n1(9.80)

and assuming that the difference between the indices of refraction is much smaller thantheir sum,

ηbe � |n2 – n1|n2 + n1(9.81)


The fractional band gap width is

�ω

ω0=

2( |n2–n1|n2+n1

)π /2

=4π

( |n2 – n1|n2 + n1

)(9.82)

9.4.4 Transfer matrix of periodic stacked layers

We are interested in finding the reflection and transmission through a periodic layeredstack consisting of N layers. Owing to the periodicity, the transfer matrices of the suc-cessive N steps must be identical so the overall transfer matrix must be equivalent to thematrix of a unit-step transfer raised to the power of N . Denote the overall ABCDmatrixasM(N�) and the individual unit-cell displacement matrices asMn(�). Then

M(N�) =M1(�) ·M2(�) ·M3(�) . . .MN(�) =M(�)N (9.83)

Now we invoke a result from the theory of matrices to write an expression forM(�)N :

M(�)N =

[AUN–1(a) –UN–2(a) BUN–1(a)

CUN–1(a) DUN–1(a) –UN–2(a)

](9.84)

where

a =12(A +D) (9.85)

and the functions U(a) are the Chebyshev polynomials of the second kind:

UN(a) =sin[(N + 1) cos–1(a)

]√1 – a2

(9.86)

But from Equations 9.70 and 9.72 we find that

cos–1(a) = K� (9.87)

and manifestly

√1 – a2 = sinK� (9.88)

so

UN(a) =sin(N + 1)K�

sinK�(9.89)

230 Metamaterials

9.4.5 Bragg reflection

Inspection of Figure 9.10 shows that the reflection coefficient from a plane wave incidentnormal to the stack must be

rN =M21

M11=

CUn–1

AUN–1 –UN–2(9.90)

and the reflectivity |r|2 is

|rN |2 = |C|2|C|2 + ( sinK�

sinNK�

)2 (9.91)

In writing Equation 9.91 we have used the fact that D∗ = A, C∗ = B, and AD – BC = 1.At the centre of the band gap K� = π + ix so that the stack reflectivity becomes

|rN |2 = |C|2|C|2 + ( sinh x

sinhNx

)2 (9.92)

(n+2)Λ

Z

XA0 B0

d

...

Λ-d

(n+1)Λ

(n–1)Λ

(n)Λ

ε1

ε2

AN

ε2

ε1

ε1

ε1

ε2

ε2

Figure 9.10 Diagram of a periodic stack of dielec-tric slabs with permittivities ε1, ε2 and thicknesses(along z) d1, d2 with period� = d1+d2. The variouspaths of transmission and reflection at the successiveinterfaces is indicated by the arrows.


As N → ∞ the second term in the denominator goes to zero and the reflectivityapproaches unity.

At the band edge K� = π . The second term in the denominator of Equation 9.91becomes indeterminate and must be evaluated using L’Hôpital’s rule. The result is

|rN |2 = |C|2|C|2 + ( 1

N

)2 (9.93)

Thus, we see that both in the middle of the band gap and at the gap edge, the reflectivityapproaches unity with increasing number of periods.

So far we have determined the reflectivity as if the incident plane wave originated inthe first layer of the first period. As a practical matter we are usually more interested inthe reflectivity of a periodic stack for a plane wave incident from the air or vacuum. Wecan find this expression from Equations 9.24, 9.25, and 9.26 by supposing that incidentplane wave, originating at –∞ in the vacuum, impinges on a very thin layer of materialε1 before entering the stack. This procedure effectively allows us to use Equation 9.36while setting the phase shift ϕ2 to zero:

r =r01 + rN1 + r01rN

(9.94)

where r01 is the vacuum-ε1 reflection coefficient.

9.4.6 Resonant tunnelling

We can write the transmittance or transmissivity from energy conservation as

|tN |2 = 1 – |rn|2 =1

1 + |C|2 ( sinNK�sinK�

)2 (9.95)

We see that in the band gap, where sinNK� and sinK� are replaced by their sinh xcounterparts, the transmittance goes to zero as N → ∞. At the band edge, however,where K� = π ,

|tN |2 = 11 + |C|2N2

(9.96)

where, from Equation 9.54:

|C| =[12

(kz2kz1

ε1

ε2–kz1kz2

ε2

ε1

)sinϕ1

](9.97)

Since |C|2 varies as sin2 ϕ1 we see that the transmittance can be unity as ϕ1 = kz1d1goes through integral multiples of π . For lossless dielectric periodic stacks we, therefore,

232 Metamaterials

must conclude that within the bandgap there can be some ‘leakage’ transmission whenthe number of unit cells is not too great, but resonant transmission is confined to theadjacent band windows. We shall see later that this conclusion does not necessarily holdwhen one of the stack components exhibits a real, negative permittivity. Figure 9.11shows the transmittance, calculated from Equation 9.95 for a periodic stack consistingof 300 nm thick alternating slabs of TiO2 and SiO2. Light is incident normal to thesurface. A band gap is clearly evident with adjacent rapidly oscillating transmittancemaxima. The results of the TMT calculation of the transmittance can be compared todirect numerical simulation using a finite difference time domain (FDTD) techniqueto solve Maxwell’s equations. The numerical simulation result for the same structure isshown in Figure 9.12.

In the band gap the incident light cannot propagate and must therefore decay expo-nentially. Figure 9.13 plots the intensity of the Ex field component, determined fromthe FDTD numerical simulation, as a function of distance into the stack along z.

0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.85 0.9 0.95 1 1.05 1.1k0 (m–1)

tran

smitt

ance

× 1071.15 1.2

Figure 9.11 Transmittance through a periodic stack, calculated fromEquation 9.95, and composed of 9 periods of alternating TiO2 and SiO2slabs 300 nm thick. The bandgap is centred k0 � 1.04× 107m–1 with�k0 � 0.1× 107m–1. Rapid oscillations adjacent to the abrupt bandgap edges converge to near-unity transmittance in the band windows oneither side of the gap.


0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

tran

smitt

ance

N=9FDTD

k0 (m–1) × 107

Figure 9.12 Transmittance through the same periodic stack as in Figure 9.11calculated by direct numerical solution of Maxwell’s equations. The finitedifference time domain (FDTD) method is used to calculate the transmittance.Slight dispersion in the indices of refraction over the frequency range is taken intoaccount in this result.

The intensity decreases in steps with each unit period and exhibits an exponential en-velope consistent with Equation 9.73. Figure 9.14 plots the propagating Bloch wave inthe stack at the band edge. As expected from Equation 9.76, the envelope indicates ahalf-wave oscillation.

9.4.7 Metal-dielectric periodic layers

So far in Section 9.4.2 we have considered only lossless dielectric periodic layers. Inthis section we extend the discussion to alternating layers of dielectric and metal, ormore generally between layers of positive and negative permittivity. We will see thatsurface plasmon modes at the dielectric-metal interfaces play a crucial role in resonant

234 Metamaterials

–1 –0.5 0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Z (m)

Inte

nsity

Ex

E-field Intensity vs Z at Band Gap Center

× 10–6

Figure 9.13 Evanescent Ex field intensity, calculated from FDTD,in the TiO2-SiO2 periodic stack at the frequency centre of the bandgap. The vertical dashed lines indicated the slab period boundaries.The horizontal dashed line indicates 1/2e of the evanescent intensitydecay from the point of incidence at z = +2.7μm. The 1/2epenetration distance is consistent with Equations 9.73 and 9.76.

tunnelling of light through this type of periodic stack. We will continue to restrict thediscussion to lossless materials in order to keep the mathematical expressions manage-able and the physical ideas lucid. Real metals, of course, are always somewhat lossy, butthe results obtained here will closely approximate reality as long as the real part of thepermittivity is much greater than the imaginary part. Figure 9.15 shows a schematicof alternating layers of metal and dielectric slabs. In this schematic the top and bottomlayers are metal followed by alternating layers of dielectric and metal. The dielectric lay-ers (d1) are usually thicker than the metal layers (d2) because the absolute value of thepermittivities of metals in the optical regime is usually greater than those of dielectrics.


–3 –2 –1 0 1 2 30

1

2

3

4

5

6

Z (m)

Ex

Inte

nsity

E-field Intensity vs Z at Band Edge

× 10–6

Figure 9.14 Propagating Ex field intensity, calculated fromFDTD, in the TiO2-SiO2 periodic stack at the band edge. Theamplitude envelope indicates a half period of the eiK� Blochwave propagation as expected from Equation 9.73 at the bandedge.

The top and bottom metal layers can be one half the thickness of the interior metal lay-ers to ensure periodic translational symmetry from top to bottom. Often a high index‘coupling’ layer is added to the top and bottom of the stack to ensure efficient sourcecoupling at Brewster’s angle to the surface plasmon index.

The key difference between dielectric periodic layers and dielectric-metal periodiclayers is the appearance of evanescent rather than propagating field components per-pendicular to the dielectric-metal interfaces. Applying matching boundary conditionsresults in surface waves that do propagate along x at the interfaces but are evanescentalong z. These waves are called ‘surface plasmon’ or ‘surface plasmon polariton’ SPPwaves. Their properties for semi-infinite slabs and a single interface have been devel-oped in Chapter 7. Here we recall the three expressions for the surface wave vectorcomponents: ksx, k

s,dz , ks,mz from Equations 7.34–7.36,

236 Metamaterials

Z

X

Y

d1

d2

Figure 9.15 Periodic stack of alternating metal and dielectric layers. The opticalentrance and exit (top and bottom) layers are metal with thickness d2/2. Light incident atthe top, with E-field vector in the X – Z plane (TM polarisation) couples to surfaceplasmon modes evanescent in z but propagating in the x – y plane at the metal-dielectricinterfaces.

ksx =√

εdεm

εd + εmk0 (9.98)

ks,dz = ± εd√εd + εm

k0 (9.99)

ks,mz = ± εm√εd + εm

k0 (9.100)

In these expressions ksx is the x component of the surface wave propagating at theinterface, ks,dz is the z component in the dielectric, and ks,mz is the z component in themetal. The terms εm, εd are the real metal and dielectric permittivities, respectively.We remember that the metal permittivities are negative, and therefore if |εm| > εd , thekz components will be pure imaginary. We write kz components as in Equations 7.39and 7.40,

ks,dz →±iκds = ±i∣∣∣∣ εd√εd + εm

∣∣∣∣ k0 (9.101)

ks,mz →±iκms = ±i∣∣∣∣ εm√εd + εm

∣∣∣∣ k0 (9.102)


The choice of sign (±) is made so that the wave amplitude decreases exponentially withincreasing distance from the interface (±z). Specifically, if there is a metal-dielectricinterface at z = 0, and the dielectric is on the +z side, then

ks,dz → iκds z > 0 (9.103)

ks,mz → –iκms z < 0 (9.104)

and the components of the surface wave projecting onto the dielectric side take on thefollowing forms,

Hs,dy (x, z, t) = H0ye–κ

ds zei(k

sxx–ωt) (9.105)

Es,dx (x, z, t) = –iκdsωε0εd

H0ye–κds zei(k

sxx–ωt) (9.106)

Es,dz (x, z, t) = –ksx

ωε0εdH0ye–κ

ds zei(k

sxx–ωt) (9.107)

Similar expressions obtain on the metallic side with the appropriate change of sign foriκms (Equation 9.104). With the change to pure imaginary arguments, Equation 9.71 forthe Bloch vector becomes [15],[16]

cosK� = cosh(kz1d1) cosh(kz2d2) +12

(kz2kz1

ε1

ε2+kz1kz2

ε2

ε1

)sinh(kz1d1) sinh(kz2d2)

(9.108)

Taking into account the form of kz1, kz2 for the evanescent z components of the sur-face plasmon wave in Equations 9.101 and 9.102, and the fact that εm = ε2 < 0, theexpression for the Bloch vector becomes

cosK� = cosh(kz1d1) cosh(kz2d2) – sinh(kz1d1) sinh(kz2d2) = cosh(kz1d1 – kz2dd2)(9.109)

This expression is valid in the limit of weak coupling of the surface plasmon modesbetween adjacent layers. Equations 9.101 and 9.102 were obtained for an isolated surfaceplasmon mode in which the metal and dielectric half-spaces extend to infinity. In thestacked layer these modes will couple to produce symmetric and antisymmetric linearcombinations that are the eigen modes of the stacked layer system. However, it appearsfrom numerical simulation that use of the weak-coupling expressions are adequate forpractical design of a resonant tunnelling stack. The cosK� expression in Equation 9.109must be equal to or less than unity for the cosine argument to be real. When kz1d1 =kz2d2, cos(K�) = 1 and K = 0, 2π , 4π , . . .. Thus, when the phases kz1d1, kz2d2 areequal, the Bloch vector is real but stationary. The phase velocity of the Bloch wave alongz is null although the light fields are linked evanescently from layer to layer. Plane wavesincident on the stack emerge from it without change of phase, due to this linked, resonanttunnelling along z. Equations 9.105–9.107 show that the plasmon waves do propagatealong the x direction, transverse to z. Figure 9.16 plots kz1d1, kz2d2, and cosK� as a

238 Metamaterials

700 720 740 760 780 800 820 840 860 880 9000.7

0.8

0.9

1

1.1

1.2

1.3

1.4

λ0 (nm)

kz1d1

kz2d2

cos(KΛ)

TiO2: d1 = 94 nmAg: d2 = 20 nm

Figure 9.16 Plots of kz1d1, kz2d2, and cosK� versus λ0, the wavelength of light incident on aperiodic stack of TiO2 layers (d1 = 95 nm) and Ag (d2 = 20 nm). The plots show that for thisparticular choice of materials and layer thicknesses, cos(K�) = 1 in a relatively narrow range ofwavelengths centred at λ0 = 788.9 nm.

function of the wavelength λ0 of incident light for a typical example of alternating layersof a silver-titanium dioxide (Ag-TiO2) periodic stack. In this particular example thedielectric layer thickness (d1) is chosen to be 94 nm and metal layer thickness (d2) is 20nm. Figures 9.17 and 9.18 show the results of an FDTD simulation of the λ0 = 785 nmplane wave incident on a five-layer stack constructed with the materials and geometryused in Figure 9.16. Figure 9.17 shows a plot of the Ex field amplitude in the X – Zplane of the periodic Ag-TiO2 stack. The plane wave is incident on the top, a 10 nmthick Ag surface at z = 684nm, and at –50.8 degrees from the normal in a slab of silicon(Si) above the stack. This Si slab permits excitation of the surface plasmon modes atBrewster’s angle. A corresponding Si slab substrate at the stack exit couples out thepropagating plane wave at z = 0nm. For this example of a Ag-TiO2 system, the zeroth-order wavelength of the surface plasmon, corresponding to an isolated, single interface,


λ0 = 789.0 nm, θ = –50.8 deg, Ex Field, AgTiO2 Stack

X-axis (nm)

Z-a

xis

(nm

)

–2000 –1500 –1000 –500 0 500 1000 1500 2000

–100

0

100

200

300

400

500

600

700

800

900

Plane Wave Source

Figure 9.17 A plane wave source located at z = 800 nm impinges on a periodic stack of alternatingAg layers (20 nm thickness) and TiO2 layers (94 nm thickness). Top Ag surface, z = 684 nm; bottomAg surface, z = 0 nm. Top and bottom Ag layers are 10 nm thick. The source and stack are embeddedin a layer of Si on top and another on the bottom that ensure efficient coupling into and out of theplasmon modes. The source wavelength λ0 = 789 nm, with E-field vector in the X – Z plane (TMpolarisation) and is incident on the top surface at an angle of –50.8 deg with respect to the stacknormal. The plot shows Ex with positive and negative amplitudes indicated by red and blue. Withinthe stack the fields are linked evanescently along z while propagating as guided waves in the +xdirection. Note that absence of phase shift along z indicating a null phase velocity of the Bloch wave.Similar plots obtain for field components Ez and Hy.

is λspp = 278nm. In the simulated stack of Figure 9.17 the surface plasmon wavelengthis 277 nm, very close to the single-interface value. Figure 9.19 shows the amplitude ofthe SPP mode along x within the TiO2 layer at z = 670nm.

A metal-dielectric periodic stack, exhibiting resonant tunnelling along z, also exhibitsthe properties of an anisotropic ‘epsilon-near-zero’ (ENZ) metamaterial [17],[18]. Ac-cording to the effective medium theory (EMT) characterising metamaterials, permittivitiesperpendicular and parallel to the optical axis are given by,

240 Metamaterials

0 100 200 300 400 500 600 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Z (nm) @ X = 0

Inte

nsity

Ex

Intensity Ex vs. Z @ X = 0

TiO2 TiO2 TiO2 TiO2 TiO2TiO2Ag Ag Ag Ag Ag

Figure 9.18 Intensity profile of the illuminated stack shown in Figure 9.17. E-field Intensity along zat x = 0 from the top surface (right) at z = 694 nm to bottom surface (left) at z = 10 nm. The profileshows how the fields are evanescently linked along z from incident (left) to exit (right) surfaces.

ε⊥ =d1ε1 + d2ε2d1 + d2

(9.110)

1ε‖

=1

d1 + d2

[d1ε1

+d2ε2

](9.111)

Figure 9.20 shows the effective medium permittivities for the Ag-TiO2 stack as a func-tion wavelength around the region where ε⊥ � 0. We see from Equations 9.108, 9.101,and 9.102 and 9.110 that the condition for a real Bloch vector, kz1d1 – kz2d2 = 0,also implies, at least in the weak-coupling limit, that ε⊥ = 0. Transmission by res-onant tunnelling through surface plasmon modes [19] is intimately connected to the‘epsilon-near-zero’ (ENZ) metamaterial [20–24] that holds promise for many novel andunexpected material properties. In particular, Engheta et al. [23] have pointed out thata point dipole above a surface of ENZ material will be subject to a repulsive force


–2000 –1500 –1000 –500 0 500 1000 1500 2000–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

X (nm) @ Z = 670 nm

Ex

SPP Ex Field vs. X @ Z = 670 nm AgTiO2 Study

λspp = 277 nm

Figure 9.19 Amplitude of the surface plasmon Ex component along the x direction within a TiO2layer at z = 670 nm (see Figure 9.17). The simulated FDTD λspp = 277 nm, is very close to thewavelength λ0 = 278 nm of a surface plasmon at a single, isolated interface.

analogous to the expulsion of a magnetic dipole from a superconductor due to theMeiss-ner effect. Figure 9.21 illustrates the analogy and Figure 9.22 plots an FDTD simulationof the D-field generated by a point dipole close to an ENZ surface. According to [23]the time-averaged vertical component of the repulsive force is given by

〈Fz〉 = –σ9

512π4cRe[εmatl – ε0εmatl + ε0

]Prad

(λ

d

)4

(9.112)

where σ = 1, 2 if the alignment of the dipole is horizontal, vertical with respect to thesurface plane, and d, λ are the distance above the plane and the wavelength, respectively.The factor Prad is the spatially averaged radiated power of a classical dipole p in freespace (see Equation 5.7 in Chapter 5) and is given by

Prad =4π3c3ε0λ4

· |p|2 (9.113)

242 Metamaterials

740 750 760 770 780 790 800 810 820 830 840−1

0

1

2

3

4

5

6

7

8

9

λ0 (nm)

ε per

p, ε p

ara

EMT calculations εperp, εpara vs. λ0

εperp

εpara

Figure 9.20 Plots of the effective medium theory (EMT) relative permittivities parallel (ε‖) andperpendicular (ε⊥) as function of wavelength for the Ag-TiO2 periodic stack characterised inFigures. 9.16–9.19. The terms ‘parallel’ and ‘perpendicular’ refer to the optical (z) axis. Theperpendicular permittivity is near zero between λ0 = 785 – 790 nm .

When εmatl → 0 the effective vertical force is repulsive. In the case of Figure 9.20 wesee that ε⊥ → 0 around λ = 785nm. This wavelength is very close to the resonancetransition of a rubidium (Rb) atom, and we can calculate the equilibrium position ofan atom subject to this levitation force and the force of gravity. Figure 9.23 shows aplot of the levitation force for the Rb atom as a function of distance from the plane ofthe anisotropic material. The equilibrium point is where this curve crosses the constantgravitational force.

9.4.8 Anisotropy in stacked layers

Returning to Figure 9.20, we emphasise in this section the anisotropy in the paralleland perpendicular permittivities. In k-space this property can be represented by thegeometry of the dispersion equation. For a lossless structure conservation conditionsrequire


Superconductor (B = 0)

ε-Near-Zero material (D = 0)

(a)

(b)

N S B

p(t)D

Figure 9.21 Top diagram (a) shows how the B-field is excluded froma superconductor and results in a strong B-field gradient just abovethe surface. The magnetic dipole is repelled by a dipole-gradient force.Bottom diagram (b) shows the analogous D-field gradient and repul-sive force resulting from an electric dipole above an ENZ material.Figure adapted from [21] and used with permission.

n2xk2x + n

2yk

2y + n

2zk

2z = n2k20 (9.114)

where we take z to be the optic axis and nx, ny, nz are the refractive indices along x, y, z,and k0 = 2π /λ0 = ω/c is, as usual, the propagation parameter in free space. We then have

εxk2x + εyk2y + εzk

2z = ε‖ε⊥k

20 (9.115)

where εx, εy = ε⊥ and εz = ε‖ denote the permittivities perpendicular and parallel to theoptic axis1. Dividing both sides by ε⊥ε‖, we can then write,

k2xε‖

+k2yε‖

+k2zε⊥

= k20 (9.116)

Equation 9.116 is conventionally called the dispersion equation for a uniaxial anisotropicmaterial. It defines a 3-D surface–the isofrequency surface. Figure 9.24 shows the differentclasses of surface defined by the dispersion equation as a function of the relative mag-nitudes and signs of the perpendicular and parallel permittivities. Clearly, our exampleAg/TiO2 stacked layer goes from ellipsoidal at wavelengths less than 785 nm to a one-sheet hyperbolic surface at wavelengths greater than 790 nm. There has recently been

1 Some authors take ε⊥, ε‖ perpendicular and parallel to the x – y plane.

244 Metamaterials

–2

7

7.5

8

8.5

9

9.5

10

–1.5

εperp = 0, εpara = 1

–1 –0.5 0X-axis (m)

Z-a

xis

(m)

Dz Dipole

0.5 1 1.5 2

–400

–300

–200

–100

100

200

300

400

0

× 10–7

× 10–7

Figure 9.22 An FDTD numerical simulation of a point dipole placed 650 nmabove an anisotropic material with εperp = 0 and εpara = 1. The contours plotthe Dz component of the displacement field. Anisotropic ENZ metamaterialscan be realised at a particular wavelength by stacked layers of dielectric andmetal of fixed geometry (see Figure 9.20).

a great deal of interest in hyperbolic metamaterials because a dipole emitting radiationinto a hyperbolic material can, in principle, couple to a much higher density of statesthan when the same dipole couples radiatively to the vacuum. We recall from Chapter 6that the density of states available for spontaneous emission into the isotropic vacuumincreases as k20 and corresponds to the surface of the isofrequency sphere. Suppose nowthat an emitting dipole is situated just above the surface of an isotropic material. Fig-ure 9.25a shows a schematic representation. At the surface the k-vector componentsparallel to the x – y plane must be continuous through the surface. Therefore, for theclosed-surface isotropic sphere, the upper limit to kx is k0 pointing somewhere in thex – y plane. In contrast, if the material isofrequency surface is an open two-sheet hyper-boloid, as shown in Figure 9.25b, any kx vector can satisfy the continuity condition atthe boundary, and therefore, in principle, the number of k-states available for emissionis greatly enhanced. This condition should give rise to a marked increase in the radiativerate of a quantum emitter. Reference to Figure 9.20 shows that for the Ag-TiO2 stackedlayer, it is ε⊥ that goes negative, and therefore, the appropriate isofrequency surface is


300 320 340 360 380 400 420 440 460 480 5000

1

2

3

4

5

6

7

8× 10–24

Distance from ENZ Surface (nm)

For

ce (

N)

ENZ Repulsion Force vs. Distance From Surface

Gravitational force on Rb atom

Figure 9.23 Levitation force < Fz > and gravitational force on a Rb atom as a function of distancefrom an anisotropic metamaterial with ε⊥ → 0.

the one-sheet hyperboloid, Fig. 9.24 (d). In this case the minimum-length k-vector is k0pointing in the x – y plane, while for all longer k-vectors the continuity condition canalways be satisfied. It is interesting to note that at the wavelength crossover point in Fig-ure 9.20, where ε⊥ = 0, the available k-vectors are also all confined to the x – y plane.Rather than a surface, the k-vectors that satisfy the continuity condition are confined toa ring in the x – y plane of radius k0. Since the total number of k-states is considerablyless than in the isotropic vacuum, one expects a marked decrease in the rate of emission.

9.4.9 Validity of effective medium theory

In this section we examine the validity of the effective medium theory used in Sec-tion 9.4.7 and in particular the use of Equations 9.110 and 9.111. The effective mediumtheory (EMT) presupposes that the wavelength of light interacting with a structuredmaterial is very long compared to the characteristic length scale of any component of

246 Metamaterials

–1–0.5

00.5

1

–1

–0.5

0

0.5

1–0.5

0

0.5

(a) (b)

(c) (d)

–11

0.5

–0.5–0.5

00.5

1

–1 –1

0

10–15

–10

–5

0

5

10

15

5–5 –10 –10

–5 0 5 10

KxKy

KxKy

0 10–15

–10

–5

0

5

10

15

5–5

–10 –10–5 0

5 10

KxKy

0

–0.8–0.6

–0.4

Kz

–0.20

0.20.40.60.8

1

Figure 9.24 (a) sphere: isotropic, ε⊥ = ε‖ (b) ellipsoid: ε⊥ > ε‖ (c) hyperbolic: two-sheet, ε‖ < 0 (d)hyperbolic: one-sheet, ε⊥ < 0.

that composite. This assumption is usually well-founded when the components are alldielectrics whose refractive indices are not greater than, say, 2.5. But in the case ofmetal-dielectric layers, even when the layer thicknesses are subwavelength, the char-acteristic plasmon waves at the interfaces may have wavelengths greatly reduced fromthe incident light. This important wavelength reduction is due to the large imaginaryterms of metal refractive indices in the visible and near infra-red spectral regions. Therange of legitimate application of the effective medium theory may therefore be greatlyreduced. However, we can compare the anisotropic permittivities determined by thesimple EMT formulas that assume constant field amplitudes within each subwavelengthlayer, to those extracted by the more accurate and precise transfer matrix theory thatnumerically matches appropriate field amplitudes at each interface.

Before describing this extraction procedure it is important to note that for somemetamaterial design goals, a useful answer can be obtained independently of the EMTassumptions. As Smith et al. [25] have pointed out, the Bloch condition for periodicstructures, Equation 9.72, is insensitive to the length scale of the individual components.


emitting dipole

z

emitting dipole

kx

kx

z

k0

k0

Figure 9.25 (a) Isofrequency sphere showing the surface to which k0 must be constrained.The limiting amplitude of any component in the x – y plane is therefore k0. (b) Isofrequencytwo-sheet hyperboloid showing the open surface to which, in principle, any k0 can couplebecause the continuity condition at the surface can always be satisfied.

The Bloch vectors associated with these structures should therefore be valid whether ornot the EMT formulas apply. In particular, in the case of alternating slabs of TiO2 andAg, if the goal is to find a condition for ENZ behaviour, Figure 9.16 shows in what wave-length region the Bloch K vector is close to zero. This is also the condition for resonanttunnelling through the stack as illustrated in Figures 9.17 and 9.18. Note that the plot ofEMT permittivities in Figure 9.20 also shows that ε⊥ � 0 in the same wavelength regionas the Bloch vector. For the ENZ property one can say that effective medium theoryyields a wavelength region that is at least consistent with a Bloch vector behaviour, it-self independent of EMT assumptions. This conclusion, however, does not mean that astack of alternating TiO2 and Ag slabs with given slab thicknesses is a good approxima-tion to a homogenous anisotropic material with ε⊥ and ε‖ as indicated in Figure 9.20. Inorder to answer that question the following extraction procedure may prove useful.

The procedure for this extraction [26] begins with the matrix expressions used for themetal layer and the dielectric layer in a periodic stack. For the time being we will considerthe individual layers to be isotropic, and the overall transmission matrix TT correspondsto a uniform material composed of the dielectric and metal slabs. Figure 9.26 shows theschematic layout used for the procedure and the matrices for the individual slabs are

Tm =

[cosϕm i cos θ√

εmsinϕm

i√εm

cos θ sinϕm cosϕm

](9.117)

with

ϕm =2πλl√εm

√1 –

εa sin2 θεm

(9.118)

248 Metamaterials

Metal εm

z

xDielectric εd

Incidentwave

tmtd

Externalmaterial-

εa

Externalmaterial-

εaS11

S21

t = td + tm

θFigure 9.26 Schematic of the metal-dielectric periodic stack structure. TheS-parameters S11,S21 correspond to net ormeasured reflection at the incident port andnet or measured transmission at the exit port,respectively, with the angle of incidence θ . Thethicknesses for the dielectric and metal aretd , tm, respectively. The relative permittivitiesof the metal, dielectric, and surroundingmaterial (usually air or glass) are εm, εd, andεa, respectively. Figure adapted from [26] andused with permission.

where the notation for the various terms are defined in Figure 9.26 and

Td =

[cosϕd i cos θ√

εdsinϕd

i√εd

cos θ sinϕd cosϕd

](9.119)

with

ϕd =2πλd√εd

√1 –

εa sin2 θεd

(9.120)

The overall transmission through the stack composed of N dielectric slabs and N + 1metal slabs is the matrix product of the matrices for the individual dielectric and metalslabs:

TT = TmN∏i=1

TdTm =[T11 T12

T21 T22

](9.121)

The next step is to write the scattering S-parameters (the measurable quantities) for the‘homogenised’ stack of slabs in terms of the transmission matrix elements [25]. For aslab of homogenous material T11 = T22 = Ts and the S matrix is symmetric:


S21 = S12 =1

Ts + 12

(ik0T12 +

T21ik0

) (9.122)

S11 = S22 =

12

(T21ik0

– ik0T12

)Ts + 1

2

(ik0T12 +

T21ik0

) (9.123)

With the S21,S11 parameters in hand we can obtain the impedance Z and thepropagation parameter in the material, k′z, from the following expressions, [25]

Z =

√(1 + S11)2 – S2

21

(1 – S11)2 – S221

(9.124)

and

k′z =1tcos–1

[1

2S21

(1 – S2

11 + S221

)+ 2πn

](9.125)

where t is the period unit thickness, t = td + tm, and n is an integer. But the S-parameterscan also be written in terms of the Fresnel reflection coefficient R. We write approxi-mate expressions for the S matrix elements, taking into account multiple reflections andtransmission from the component slabs and dropping terms with R3 or higher powersof R:

S11 � R(1 – ei2k′zt)

1 – R2e2ik′zt(9.126)

S21 �(1 – R2

)eik′zt

1 – R2e2ik′zt(9.127)

The reflectivity R is determined from these expressions because S11,S21, k′z have beenpreviously determined by Equations 9.122, 9.123, and 9.125. Furthermore, we can writethe Fresnel reflection and transmission coefficients, R,T of the homogenised material bytangential field component matching (with TM polarisation) at the incident interface,using Equations 7.6–7.8:

Eix – REix = TEtx (9.128)

k0 cos θ√εa

ε0εa– R

k0 cos θ√εa

ε0εa= T

k′zε0εx

(9.129)

T =k0εxk′z√εa

cos θ(1 – R) (9.130)

Hiy + RH

iy = THt

y (9.131)

1 + R = T (9.132)

250 Metamaterials

Eliminating T from Equations 9.130 and 9.132:

R =

k0k′z

εx√εacos θ – 1

k0k′z

εx√εacos θ + 1

=1 – k′z

k0

√εaεx

( 1cos θ

)1 + k′z

k0

√εaεx

( 1cos θ

) (9.133)

In writing Equation 9.130 we have labelled εx explicitly in anticipation of an anisotropybetween εx and εz in a homogenised but anisotropic material. The reflectivity can bewritten in terms of the ‘normalised impedance’ as

R =1 – Z

Z0

1 + ZZ0

(9.134)

from which we identify that

ZZ0

=k′zk0

√εa

εx

(1

cos θ

)(9.135)

The impedance Z0 corresponds to the impedance of the external material (usually airor glass) with relative permittivity εa and Z is the impedance of the homogenised stack.We now suppose that our layered stack exhibits anisotropy in the permittivities with εx =εy �= εz. Figure 9.27 shows how the layered stack of isotropic, alternating material slabs

Homogeneousand anisotropic

medium

z z

x x

Externalmaterial-

εa

Externalmaterial-

εa

Externalmaterial-

εa

Externalmaterial-

εaS11 S11

S21 S21

Dielectric εd

θ

Incidentwave

θ

Incidentwave

Metal εm

tmtd

t = td + tm

εx = εy, εz

Figure 9.27 The stacked layers of periodic, alternating dielectric and metal slabs are modelled as ahomogenous, anisotropic uniaxial material with optical axis along z. The thickness t = td + tmcorresponds to the thickness of a unit cell in the stacked layer. Figure adapted from [26] and usedwith permission.


can be modelled as a homogenous anisotropic material of thickness t. Finally, we obtainfrom Equation 9.135 a determination of the effective relative permittivity perpendicularto the optical axis as a function of the unit cell thickness:

εx =Z0

Zk′z(t)k0

(1

cos θ

)(9.136)

Then, using Equation 9.115, where evidently ε⊥ = εx and ε‖ = εz,

εz(t) =εx(t)k2xεxk20 – k

′2z

(9.137)

740 750 760 770 780 790 800 810 820 830 840–1

0

1

2

3

4

5

6

7

8

9

λ0 (nm)

εpara TMT

εperp TMT

εperp EMT

εpara EMTtd = 94 nm

tm = 20 nm

εpara, εperp calculated with EMT, TMT vs.λ0

ε par

a, ε p

erp

Figure 9.28 Plots of ε‖, ε⊥ calculated according to TMT extraction using Equations 9.136 and9.137, and EMT using Equations 9.110 and 9.111 vs λ0 at normal incidence. The TMT extractionis called the complete parameter retrieval (CPR) approach in [27]. The EMT plots are the same as inFigure 9.20. The ‘filling factor’ parameter ρ is the metal thickness fraction to the total unit thickness,tm/(td + tm). The value ρ = 0.2 corresponds approximately to the parameters tm = 20 nm andtd = 94 nm used in the example of Sections 9.4.7 and 9.4.8. Figure data from [27] and used withpermission.

252 Metamaterials

Figures 9.28 and 9.29 show plots of ε‖, ε⊥, calculated according to Equations 9.136 and9.137 as a function of the wavelength λ0 in the vicinity where the effective medium for-mulas predict ε⊥ � 0. The plots compare the results of the TMT procedure to the EMTcalculations. The results of Figure 9.28 show that the EMT approximation is not reliablewith slab thicknesses of 20 nm and 94 nm for Ag and TiO2, respectively; but by redu-cing both thicknesses by a factor of 3, Figure 9.29 shows that good agreement betweenEMT and TMT can be obtained. Note that the geometry for ‘good agreement’ is atthe limit of standard metal vapour deposition technology for metal layer thickness. Weconclude, therefore, that a stack of alternating slabs of dielectric and metal can indeedmimic a homogenous, anisotropic material but the EMT approximation does not pro-vide a reliable guide for design purposes even when the slab geometries are nominally

740 750 760 770 780 790 800 810 820 830 840−1

0

1

2

3

4

5

6

7

8

9

λ0 (nm)

ε par

a, ε p

erp

εpara, εperp calculated with EMT, TMT vs. λ0

td = 31 nm

tm = 7 nm

εpara TMT

εperp TMT

εperp EMT

εpara EMT

Figure 9.29 Plots of ε‖, ε⊥, calculated according to TMT extraction using Equations 9.136 and9.137, and EMT using Equations 9.110 and 9.111 vs λ0 at normal incidence. The TMT extractionis called the complete parameter retrieval (CPR) approach in [27]. The EMT plots are the same as inFigure 9.20. The filling factor ρ = 0.2 is the same as in Figure 9.28 but tm = 7 nm and td = 31 nm.Figure data from [27] and used with permission.

Bibliography 253

well into the ‘subwavelength’ regime. The TMT procedure should be as reliable as theconstituent material properties since it is based only on geometry of the stack, the indi-ces of refraction of the dielectric and metal components, and numerical field matchingat the layer interfaces.

9.5 Summary

This chapter begins with a brief summary of the definitions and properties of left-handed materials, then proceeds to a discussion of negative-index waveguides in thesubwavelength régime. The focus then turns to reflection and transmission in stackedlayers: first of alternating dielectrics and then of periodic dielectric-metal 2-D structures.Bloch waves and bandgaps appear as a consequence of the periodicity. In particular, it isshown that for judicious choices of material and geometry in a periodic dielectric-metalstack, a Bloch vector of K = 0 appears over a narrow range of wavelength. This Blochvector corresponds to resonant evanescent-wave tunnelling through the stack. The re-sult is a near-unity transmission, independent of the number of periods in the stack. Itis also shown that in addition to resonant tunnelling, the dielectric-metal periodic stackalso exhibits anisotropy with the permittivity parallel to the optical axis different fromthe permittivity of the plane perpendicular to the optical axis. The resonant tunnellingregion also corresponds to an ‘epsilon-near-zero’ (ENZ) condition for the permittivityperpendicular to the optical axis. The possibility of levitating cold atoms (radiating at theappropriate frequency) above an ENZ material is briefly considered. The chapter endswith a critical examination of the ‘effective medium theory’ (EMT) for determiningeffective anisotropies in metal-dielectric stacked layers.

9.6 Bibliography

[1] V. E. Veselago and E. E. Narimanov, The left hand of brightness: past, present andfuture of negative index materials, Nat Mater vol 5, pp. 759–762 (2006).

[2] P. Yeh, Optical Waves in Layered Media, Wiley-Interscience, Hoboken, New Jersey(2005).

[3] V. G. Veselago, The Electrodynamics of Substances with Simultaneously NegativeValues of ε and μ, Sov Phys Uspekhi vol 10, pp. 509–514 (1968).

[4] J. B. Pendry, Negative Refraction Makes a Perfect Lens, Phys Rev Lett vol 85, pp.3966–3969 (2000).

[5] J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, Planar metal plasmonwaveguides: frequency-dependent dispersion, propagation, localization, and loss beyondthe free electron model, Phys Rev B: Condens Matter vol 72, p. 075,405 (2005).

[6] J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, Plasmon slot wave-guides: Towards chip-scale propagation with subwavelength-scale localization, Phys RevB: Condens Matter vol 73, p. 035,407 (2006).

254 Metamaterials

[7] H. J. Lezec, J. A. Dionne, and H. A. Atwater, Negative Refraction at VisibleFrequencies, Science vol 316, pp. 430–432 (2007).

[8] J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, Are negative index ma-terials achievable with surface plasmon waveguides? A case study of three plasmonicgeometries, Opt. Express vol 16, pp. 19,001–19,017 (2008).

[9] D. M. Pozar, Microwave Engineering, 3rd edition John Wiley & Sons, Inc. The lefthand of brightness: past, present and future of negative index materials, Nat Mater vol5, pp. Hoboken, New Jersey (2005).

[10] N. W. Ashcroft and N. D. Mermin, Solid State Physics The Electrodynamics of Sub-stances with Simultaneously Negative Values of ε and μ, Sov Phys Uspekhi vol 10, pp.Thomson Learning, London (1976).

[11] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals Negative Re-fraction Makes a Perfect Lens, Phys Rev Lett vol 85, pp. Princeton University Press,Princeton (1995).

[12] J. Chilwell and I. Hodgkinson, Thin-films field-transfer matrix theory of planar multi-layer waveguides and reflection from prism-loaded waveguides, J Opt Soc Am A vol 1,pp. 742–753 (1984).

[13] K. H. Schlereth and M. Tacke, The Complex Propagation Constant of MultilayerWaveguides: An Algorithm for a Personal Computer, IEEE J. Quantum Electron vol26, pp. 627–630 (1990).

[14] P. S. Russel, T. A. Birks, and F. D. Lloyd-Lucas, Photonic Bloch Waves and PhotonicBand Gaps, pp. 585–633. Planar metal plasmon waveguides: frequency-dependent dis-persion, propagation, localization, and loss beyond the free electron model, Phys Rev B:Condens Matter vol 72, p. Plenum Press, New York (1995).

[15] S. Feng, J. M. Elson, and P. L. Overfelt,Transparent photonic band in metallodielectricnanostructures, Phys Rev Lett vol 72, p. 085,117–1–6 (2005).

[16] S. Feng, J. M. Elson, and P. L. Overfelt,Optical properties of multilayer metaldielectricnanofilms with all-evanescent modes, Opt Express vol 13, pp. 4113–4124 (2005).

[17] A. Alù and N. Engheta, Optical nanotransmission lines: synthesis of planar lefthandedmetamaterials in the infrared and visible regimes, J Opt Soc Am B: Opt Phys vol 23,pp.571–583 (2006).

[18] M. Silveirinha and N. Engheta, Tunneling of Electromagnetic Energy through Sub-wavelength Channels and Bends using ε -Near -Zero Materials, Phys Rev Lett vol 97,pp. 157, 403–1–4 (2006).

[19] S. Tomita, T. Yokoyama, H. Yanagi, B. Wood, J. B. Pendry, M. Fujii, andS. Hayashi, Resonant photon tunneling via surface plasmon polaritons through one-dimensional metal-dielectric metamaterials, Opt Express vol 16, pp. 9942–9950(2008).

[20] A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, Epsilon-near-zerometamaterials and electromagnetic sources: Tailoring the radiation phase pattern, PhysRev B: Condens Matter vol 75, pp. 155,410 (2007).

[21] Y. Li andN. Engheta, Supercoupling of surface waves with ε -near -zero metastructures,Phys Rev B: Condens Matter vol 90, pp. 201,107 (2014).

Bibliography 255

[22] R. Maas, J. Parsons, N. Engheta, and A. Polman, Experimental realization of anepsilon-near-zero metamaterial at visible wavelengths, Nat Photonics vol 7, pp. 907–912 (2013).

[23] F. J. Rodríguez-Fortuño, A. Vakil, and N. Engheta, Electric Levitation Using ε -Near - Zero Metamaterials, Phys Rev Lett vol 112, pp. 033,902 (2014).

[24] F. J. Rodriguez-Fortuño, A. Vakil, and N. Engheta, Electric Levitation Using ε -Near - Zero Metamaterials: Supplementary Information, Phys Rev Lett vol 112, pp.033,902 (2014).

[25] D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, Electromagnetic param-eter retrieval from inhomogenous metamaterials, Phys Rev B: Condens Matter vol 71,pp. 036,617 (2005).

[26] B.-H. Borges and A. F. Mota,Metal-dielectric layered media and hyperbolic metama-terials (2015). Presented at the Workshop on Strongly Coupled Field Theories forCondensed Matter and Quantum Information Theory, Natal, Brazil.

[27] A. F. Mota, A. Matins, J. Weiner, F. L. Teixeira, and B.-H. V. Borges, Constitu-tive parameter retrieval for uniaxial metamaterials with spatial dispersion, Phys RevB 115410 (2016).

10

Momentum in Fields and Matter

The argument has not, it is true, been carried on at high volume, but the list ofdisputants is very distinguished.1

10.1 Introduction

In Chapter 2, Section 2.5.3, we developed the idea that Poynting’s vector S could beinterpreted as the flow of electromagnetic energy across a closed surface around somepoint r. For fields propagating through spaces (vacuum or material) or on surfaces, thePoynting vector describes the field energy flux (S.I. units of Wattsm–2) and is identifiedwith the cross product between the electric and magnetic fields,

S = E×H (10.1)

Using Maxwell’s equations, Equations 2.25–2.28, together with Equation 10.1, it can beshown fairly straightforwardly how energy flows out of the enclosed volume and interactswith free and bound currents that may be present:

∇ · S +∂

∂t

(12ε0E · E +

12μ0H ·H

)+ E · Jfree + E · ∂P

∂t+H · ∂M

∂t= 0 (10.2)

In the same sense that Equation 2.35 provides a charge-current continuity across a closedsurface, Equation 10.2, Poynting’s theorem in differential form, states that the energyflux across a closed surface must be equal to the sum of the time rate of energy changeresiding in the fields, the free current sources, and the bound current sources. Clearly,the two terms in parentheses are the electric and magnetic field energies. A positive flowof energy outwards from a point implies a decrease in the field energies at that point.The E · Jfree term is interpreted as the work done by an electric field E on a free currentsource J at the point in question. If the Ohm’s law constitutive relation holds, J = σE,(σ being the conductivity), then this term can be written as J2/σ , which indicates a ‘Joule

1 Reported in [1] and attributed to E. I. Blount.


Einstein Box thought experiment 257

heating’ or irreversible ohmic resistive power dissipation. The last two terms are alsofield-current interactions, but here they involve the bound currents of polarisation andmagnetisation, ∂P/∂t and ∂M/∂t, respectively, rather than the free current Jfree.

If Poynting’s theorem states how energy can flow between current sources and fields inspace and in matter, one might ask if momentum and momentum flow could not also beassociated with electromagnetic fields. Newton’s second law asserts that a force appliedto a body is equal to the rate of change of momentum transferred to or from that body.When we apply Newton’s second law of motion, we usually think of a scattering eventwhere one body carrying momentum impinges on another, exerting a force betweenthe two bodies and altering their motion. We normally consider the momentum transferto be instantaneous upon contact. But suppose the two bodies are electrically chargedand are separated by a great distance. Motion of the first body must be transmittedto the second by the Coulomb field, but this transmission cannot be faster than thespeed of light. Therefore, the field itself must somehow receive and transmit momentumbetween the two bodies. From a less classical point of view one might imagine a two-level atom at rest in the upper state that emits a quantum of light. It is well knownfrom cold-atom experiments that the atom suffers a backward recoil motion from thedirection of light emission. Clearly, the atom has assumed a mechanical momentumequal to the product of its mass and velocity. After propagating some distance the emittedlight encounters a second two-level atom at rest in the lower state. The atom absorbs thelight and receives a forward impulsive recoil. The forward recoiling atom assumes amomentum equal to the product of its mass and velocity, which is equal in magnitudebut opposite in sign from the initial backward recoil. Conservation of linear momentumrequires that the momentum assumed by the second atom must have been transportedfrom the first by the propagating light quantum or ‘photon’. Assuming for the momentthen, that a quantity of linear momentum might be associated with a temporal lightpulse, a ‘thought experiment’ attributed to Einstein suggests what the relation might bebetween the exchange of energy and exchange of momentum.

10.2 Einstein Box thought experiment

Consider a pulsed light emitter, for example a pulsed laser, mounted on a frictionlessplatform. Suppose that on the same platform a light receiver is positioned to receive thelight pulses. How is momentum in the light pulse transferred from emitter to receiverin such a way as to be consistent with the principles of momentum conservation andcentre-of-mass conservation? The situation of the Einstein Box experiment is shownin Figure 10.1. The principle of centre-of-mass conservation means that if no externalforces act on a system, the velocity of the centre-of-mass remains constant no matterhow momentum and energy may be rearranged internally. The velocity of the centre-of-mass can of course include vcm = 0, in which case the coordinates of the centre-of-masspoint remain unchanged. In Figure 10.1 we see that prior to pulse emission,

zcm =meze +mrzrme +mr

=mrD

me +mr(10.3)

258 Momentum in Fields and Matter

emitter

frictionless surface

me receiver mr

z

D

z = 0 z = Dzcm =Dmr

me + mr

pem

ve vr

Figure 10.1 The Einstein Box thought experiment. An emitter of mass me sendsa light pulse to a receiver of mr, separated from the emitter by a distance of Dalong z. Conservation of linear momentum requires that the emitter recoilbackwards with momentum –pem and the receiver recoils forwards withmomentum pem after the pulse time-of-flight D/c. Since no external forces act onthe system, the centre-of-mass must remain unchanged by the internal momentumtransfer.

where me,mr are the masses of the emitter and receiver, ze, zr their coordinates alongz, and where initially ze = 0 and zr = D. At the initial time t = 0, both emitter and re-ceiver are at rest so the momentum of the system is zero. Now the source emits a pulse,sufficiently long that we can consider it essentially monochromatic, but sufficiently shortsuch that pulse length is much less than the distance separating emitter and receiver. Ifthe pulse carries momentum, we expect an initial recoil in the emitter so that as soonas the pulse is in flight, carrying momentum forward, the emitter moves backwards,carrying equal and opposite momentum. When the pulse arrives at the receiver, its mo-mentum is absorbed, and the receiver will recoil forwards. At all times after the transfer,the emitter will move backwards with constant velocity ve and the receiver will moveforwards with constant velocity vr so that their momenta will sum to zero:

–meve +mrvr = 0 (10.4)

Furthermore, the centre-of-mass coordinate before and after the transfer must remainconstant. After the pulse transfer, for times t > D/c, we have[(

me – Ec2

)ze(t) +

(mr + E

c2

)D +

(mr + E

c2

)zr(t)

]me +mr

=mrD

me +mr(10.5)

where the left-hand side of Equation 10.5 expresses the centre-of-mass coordinate afterthe pulse transfer and the right-hand side is the same coordinate at t = 0. We have also,

Balazs thought experiment 259

following Einstein, identified a ‘mass’ with the light pulse energy E such that E = memc2,where mem is the ‘electromagnetic’ mass and c is the speed of light in vacuum. Now, themomentum acquired by the emitter is –pem, the momentum of the light pulse, and thevelocity of the emitter after recoil is

ve =–pem(

me – E /c2) and ze(t) = vet (10.6)

Similarly for the receiver,

vr =pem(

me + E /c2) and zr(t) = vr (t –D/c) (10.7)

Substituting Equations 10.6 and 10.7 into Equation 10.5 and solving for pem results in

pem =E

c(10.8)

We see that the momentum associated with the light pulse must be its energy dividedby c. But we have already postulated that the energy flux of light must be carried bythe Poynting vector S. From the S.I. units of energy flux (Wattsm–2), it is clear thatthe energy density in the pulse must be E /V = S/c, where V is the pulse volume, andtherefore, the electromagnetic momentum density pem/V is

pemV

=E /Vc

z =Sc2

(10.9)

Of course, this thought experiment is just suggestive since the light pulse is collimated(no spatial divergence), travels through free space, undergoes no reflection, and isperfectly absorbed. What would the momentum look like if it travelled through a non-dispersive, lossless material medium with refractive index greater than unity: glass, forexample? Another thought experiment due to Balazs can help answer this question.

10.3 Balazs thought experiment

The Balazs thought experiment [2] illustrates how the centre-of-mass principle can beused to determine the momentum carried into an object from a light pulse. Again sup-pose we have a pulsed light source emitting a quasi-monochromatic pulse, the lengthof which is short compared to the spatial dimension of its overall transit. And again letus associate a mass m with the pulse, following the Einstein mass-energy equivalenceformula, E = mc2. Figure 10.2 illustrates the two situations. In the first, a light pulse isemitted and travels a distance D = ct through space with velocity c in time t. In the sec-ond, the light pulse enters, without reflection, a material slab of mass M with a lossless,real refractive index n > 1 and propagates the length of the slab in a time ts. The lightpulse group velocity vg within the slab is less than in vacuum, and so for the propagation


light pulsesource

n>1

D = ct

zz = 0

M, vsvglight pulsesource

L = vgts

frictionless surface

Figure 10.2 The Balazs though experiment consists of two situations: in thefirst a quasi-monochromatic light pulse is emitted and travels through space adistance D in time t. In the second situation the pulse travels through a losslesstransparent slab of mass M, refractive index n, and length L. The group velocityof the pulse within the slab is vg = c/n. Since no outside forces act on the system,the centre-of-mass coordinate must remain invariant between the two situations.This requirement implies that the slab must move forward some distance tocompensate for the slower pulse speed vg within the slab.

time t, the distance travelled must necessarily be less. However, whether the pulse travelsin free space or whether it travels through the slab, no external forces act on the system,and so the centre-of-mass coordinate at time t must be the same in both cases. Clearly,the centre-of-mass principle requires that the slab displaces forwards some distance tocompensate for the lower group velocity of the pulse within the slab. The principal ques-tions are: what is the momentum of the light pulse within the slab, pcms , and what is themomentum of the slab, Ps? The centre-of-mass in the first case is simply given by

zcm =m

m +MD =

mm +M

ct (10.10)

In the second case we have

zcm =( mm +M

)vgts +

(M

m +M

)vsts +

( mm +M

)c(t – ts) (10.11)

Setting the expressions for zcm in Equations 10.10 and 10.11 equal, we find that

mc = mvg +Mvs (10.12)


and since Ps =Mvs = m(c – vg), we have for the momentum of the slab,

Ps =E

c

(1 –

1n

)(10.13)

The total momentum is just the momentum of the light pulse in the first case, which wefound in the Einstein Box thought experiment to be pem = E /c. Therefore, the light pulsemomentum in the slab pems is

pems =E

c–[E

c

(1 –

1n

)]=

E

c

(1n

)=pemn

(10.14)

The momentum of the light pulse in the slab is just the light momentum in free spacedivided by the slab refractive index. This momentum is called the Abraham momentum.

From the momentum of the slab Ps we can deduce its displacement �z:

�z =PsMts (10.15)

and from Figure 10.2 we see that ts = L/vg. Therefore, we can write

�z =E /cM

(1 –

1n

)Lvg

=E /c2

M(n – 1)L (10.16)

Even though the Abraham momentum is reduced in the material slab, compared tothe same momentum packet in free space, by a factor of 1/n, the momentum density isthe same. The pulse volume in space is Act, where A is the pulse cross-sectional area andct the pulse length along z. In the slab the volume is compressed to Act/n so

pems

Vs=

pem

V= gA =

E×Hc2

=Sc2

(10.17)

where gA denotes the Abraham momentum density. From these two thought experi-ments we can tentatively postulate that the momentum density is everywhere given bythe Poynting vector divided by the square of the speed of light in vacuum.

10.3.1 Abraham momentum and Minkowski momentum

However, the postulate is tentative because the momentum density can also be plausiblyexpressed as

gM =1

ε0μ0

D×Bc2

= D×B (10.18)


This formulation is called the Minkowski momentum density, and the original mo-tivation for the Balazs thought experiment was to distinguish the consequences forAbraham momentum transferred to material bodies compared to that of Minkowski.Although the Minkowski and Abraham expressions are equivalent in free space, they arenot equivalent as the vector fields pass into the slab. The tangential field components ofE and H are continuous at the vacuum-slab interface. The total Abraham momentumof the pulse inside the slab is therefore

pemAs = gAVs = gAVn

(10.19)

in agreement with Equation 10.3. The tangential components of the D and B fields,however, are not continuous at the interface. In general for nonmagnetic materials wehave at the interface

D‖(free space) = ε0E‖,D‖(slab) = εE‖ and B‖(free space) = μ0H‖,B‖(slab) = μH‖(10.20)

where the ‘parallel’ subscripts indicate tangential components and ε, μ are the permit-tivity and permeability of the slab. In our thought experiment the slab is just a piece ofglass, so μ0 = μ, but ε > ε0. Remembering that n2 = ε, we find the Minkowski totalmomentum of the pulse inside the slab:

pemMs = gMVs = n2gAVs = ngAV = npem (10.21)

The Minkowski momentum of the light pulse inside the slab is greater than the momen-tum of the light pulse in free space by a factor of n. In order to conserve momentum andthe centre-of-mass coordinate, the slab would have to move backwards:

�z =E /c2

Mn(1 – n)L (10.22)

Since a collimated quasi-monochromatic light pulse cannot exert a ‘tractor-beam’, nega-tive light pressure force on a material body (see, however, [3]), the Abraham momentumappears to be the right choice.

But is it? Suppose we conceive another very simple thought experiment. A quasi-monochromatic plane-wave pulse propagating in a lossless medium with refractive indexn =√ε impinges on a perfect mirror embedded in the medium. What is the momentum

imparted to the mirror? Figure 10.3 illustrates the situation. The conservation of linearmomentum demands that the change in momentum of the light pulse before and afterreflection must be equal to the change in momentum of the reflector. Also, by New-ton’s second law, a change in the momentum of a body is equal to force applied to it.Therefore, a calculation of the Lorentz force at the reflector surface must be equal to thechange in momentum. We write the Lorentz force as

F = σE + Js ×B (10.23)


x

light pulse source

perfect reflectorz

refractive index = n

y

Ex

Ex

–kzkz

ByBy

Figure 10.3 Quasi-monochromatic light pulse im-pinges on a perfect reflector at normal incidence ina medium with refractive index n. Note that the re-flected E-field is π out of phase with the incidentE-field and that the incident and reflected B-fieldsare in phase. The net E-field at the surface is thereforenull and the B-field amplitude at the surface is twicethat of the incident B-field.

where σ is the surface charge density, Js the surface current density in the x – y plane,and E,B the electric field and magnetic induction field at the surface. The reflector is aperfect conductor so the amplitude of the reflected E-field changes sign at the surface.The sum of the incident and reflected E-fields is therefore null at the surface, and no fieldpenetrates into the bulk of the reflector . The B-field, however, does not change sign, andthe net B-field at the surface is twice the amplitude of Bi, the incident B-field. From theMaxwell–Ampère law, integrating along z through the interface, we find Jsx = Hy. Thenthe Lorentz force normal to the surface, averaged over an optical cycle is

Fz =12

[Jsx · B∗y

]=

12μ0[Hy · 2Hy

]= εε0E2

x (10.24)

Now we compare this Lorentz-force result to the change in the Minkowski momen-tum of the light pulse upon reflection. Suppose the incident pulse carries Minkowskimomentum density gM = D×B. Taking the absolute value and averaging over an opticalcycle we have

gM =12

[Dx · B∗y

]=

12εε0μ0ExHy =

12εε0

E2x

v(10.25)

where in the right-most term v = c/n. At the surface the momentum reverses directionbut maintains the same magnitude. Therefore, the change in momentum with respect toz evaluated at the surface is

dgMdz

= 2[12εε0

E2x

v

](10.26)


and the time rate of change of momentum is

dgMdt

=dgMdz· dzdt

= 2[12εε0

E2x

v

]· v = εε0E2

x (10.27)

The expression for the Abraham momentum density, averaged over an optical cycle, is

gA =12

[Ex ·H∗yc2

](10.28)

and following through the same reasoning as for the Minkowski expression, we find theAbraham force density applied to the reflector is

dgAdt

= ε0E20 (10.29)

We see that the force calculated from the Lorentz expression, Equation 10.24, is equalto the time rate of change of the Minkowski momentum, not the Abraham momentum.

Both the Balazs and the reflection thought experiment seem to exhibit impeccablelogic but each leads to a different expression for the momentum transfer. So what isthe correct answer? A number of real experiments [4–6] have attempted to decide theissue, but it turns out that it is extremely difficult to design and execute an experimentwith results that can be interpreted in only one way. In fact, to this day there is no wide-spread consensus, despite a number of proposals, to resolve the Abraham–Minkowskicontroversy [7–9]. To make matters worse there is a simmering, related controversy con-cerning the interpretation of ‘hidden momentum’ in magnetic materials (materials withmagnetisation field M) in the presence of charged currents. It turns out that invokingthe Minkowski expression leads to an extra term in the momentum content that mustbe reconciled with momentum conservation. We will discuss two of the key experimentsand their varying interpretations in Section 10.4.11.

10.4 Field equations and force laws

In order to frame these issues as clearly as possible, we digress to a general, but summary,discussion of the relation between Maxwell’s field equations, energy-momentum flux,and force laws. Although this discussion will not settle these controversies, the hope isthat it will lead to a deeper understanding of how the macroscopic field equations informour interpretation of electromagnetic interaction within ponderable media.

10.4.1 Maxwell’s fields in matter

In this section we follow an approach due to Mansuripur [10, 11]. As a point of depart-ure, we write down again Maxwell’s equations appropriate to electromagnetic fields inmatter:

Field equations and force laws 265

∇ ·D = ρfree (10.30)

∇ ·B = 0 (10.31)

∇ × E = –∂B∂t

(10.32)

∇ ×H = J free +∂Ddt

(10.33)

where, in polarisable, magnetic matter, D and B are related to E and H by

D = ε0E + P (10.34)

B = μ0H +M (10.35)

and ρfree and J free are the free charge density and free current density, respectively.2

10.4.2 Electric dipole or ‘Lorentz’ Model

As written, Equations 10.30–10.33, express the field relations using all four D,B,E,H;but, by invoking Equations 10.34 and 10.35, we can rewrite Maxwell’s equations in termsof E,B and the two ‘matter’ fields, P, M. The result is

ε0∇ · E = ρfree – ∇ · P (10.36)

∇ ×B = μ0

(J free +

∂P∂t

+1μ0

∇ ×M)+ ε0μ0

∂E∂t

(10.37)

∇ × E = –∂B∂t

(10.38)

∇ ·B = 0 (10.39)

From Equation 10.36 we interpret the term –∇ · P as the ‘bound charge’,

ρbound = –∇ · P (10.40)

and from Equation 10.37 we interpret the terms ∂P/∂t + 1/μ0(∇ × M) as the ‘boundcurrent’:

Jbound =∂P∂t

+1μ0

∇ ×M (10.41)

The bound charge density is the negative divergence of the polarisation field and leadsus to a material model in which the medium consists of microscopic dipoles, small withrespect to the wavelength of light, but greater than atomic dimensions. The polarisationfield itself has units corresponding to dipole density. If the dipoles are oriented within

2 Note that the definition in Equation 10.35 is not universal. Many authors use the definitionB = μ0(H+M)that has the advantage of resulting in a bound current term Jbound = ∇×M without the annoying 1/μ0 factor,but has the disadvantage of an annoying lack of symmetry with Equation 10.34.


a given volume of the material then by Gauss’s law the integral of the divergence of thepolarisation field is equal to the integral of the polarisation itself over the surface of thevolume:

–∫

∇ · P dV = –∫P · dσ =

∫ρbound dV = qbound (10.42)

where dσ is the surface element. If the dipoles are randomly oriented, this surface inte-gral will evaluate to zero. Figure 10.4 shows how the polarisation field can give rise tobound charge.

Figure 10.4 Material consisting of microscopic dipoles, the length of which is much shorterthan optical wavelengths but much longer than the atomic scale. The dipoles are arranged sothat the net divergence is non-zero. Note that within the volume the positive and negative endsof the dipoles cancel the charged ends of their neighbours, but outward from the surface there areno neighbours and the negative integral of the charge density over the surface is equal to the totalbound charge at the volume centre.


(a)

(b)

Figure 10.5 (a) Bound current in a polarisation wave built up from microscopic electric dipoleelements. Regions of positive charge density travel in the opposite direction to regions of negativecharge density to assure the continuity conditions of current and charge. (b) Bound current inmicroscopic Amperian current loops that give rise to individual magnetic moments (horizontalarrows). The density and orientation of the Amperian current loops produce the net, macroscopicmagnetisationM.

The bound current consists of two terms: the first term, dP/dt, can be thought ofas an electric polarisation current resulting from a polarisation wave travelling throughthe material. We imagine the second term, proportional to the curl of the magnetisa-tion, to consist of microscopic loops of current producing magnetic dipole moments.The units of M in the SI system is the same as B, the magnetic induction field,Wm–2 or N/(m·A). These current loops are often called ‘Amperian current loops’.Figure 10.5 shows how the microscopic-dipoles model of P and M give rise to boundcurrents.

This picture of P and M arising from microscopic electric dipoles and Amperiancurrent loops can be termed the ‘Lorentz model’ [11] because a generalised Lorentzforce FL can be written with a total charge density term and a total current density term:

FL = (ρfree + ρbound)E + (J free + Jbound)×B (10.43)


10.4.3 Magnetic dipole model

Another way to rearrange Equations 10.30–10.33 is to use Equation 10.34 and Equa-tion 10.35 to eliminate D and B. The result is

ε0∇ · E = ρfree – ∇ · P (10.44)

∇ ×H =(J free +

∂P∂t

)+ ε0

∂E∂t

(10.45)

∇ × E = –∂M∂t

– μ0∂H∂t

(10.46)

μ0∇ ·H = –∇ ·M (10.47)

We see from Equation 10.44 and 10.45 that a bound electric charge and current can stillbe interpreted in terms of microscopic electric dipoles, but Equations 10.46 and 10.47introduce a bound magnetic current, ∂M/∂t, and a bound magnetic charge, –∇ ·M, asif magnetic matter was composed of microscopic permanent magnetic dipoles. Notethe similarity between Equation 10.44 and Equation 10.47. Free magnetic monopolesare absent from Equation 10.47, but the negative divergence of the magnetisation fieldsuggests a bound magnetic charge, ρmag = –∇ ·M, arising from fixed magnetic dipolesanalogous to the picture in Figure 10.4. The Amperian current loops are no longer inevidence and have been replaced by these microscopic permanent magnetic dipoles.Both the Lorentz model and the magnetic dipole model are simply two different ways towrite Maxwell’s equations.

These ball-and-stick models of material composition are not to be taken literally.The components of matter are assembled at the atomic and molecular level and mustbe described by quantum mechanics. The models simply give us a way to visualisemicroscopic origins for the macroscopic polarisation and magnetisation fields. Fromthe Lorentz and magnetic-dipole formal rearrangements of Maxwell’s equations we canwrite expressions for the energy and momentum flux between fields and matter.

10.4.4 Energy conservation—Lorentz model

The well-known expression for charge conservation

∇ · J + ∂ρ

∂t= 0 (10.48)

states that the positive divergence of the current density from a point in space is equal tothe time rate of charge density decrease from that point. We seek something similar forenergy flowing between fields and matter:

∇ · S +∂E

∂t= 0 (10.49)


where S is the energy flux (J s–1 m–2) and ∇ ·S is the time rate of energy passing througha unit of area normal to the flux. The second term in Equation 10.49 is the time rate ofchange of energy density corresponding to the spatial flux. The conservation of energyexpression states that the spatial energy flux out of an enclosing volume must be equal tothe rate of decrease in energy density within that volume. A field divergence expressioncan be obtained from Equations 10.37 and 10.38 by operating on the first withE· and thesecond with B· and then subtracting Equation 10.37 from Equation 10.38. The result is

B · (∇ × E) – E · (∇ ×B) = B ·[–∂B∂t

]– E ·

[μ0

(J free +

∂P∂t

+1μ0

∇ ×M)+ ε0μ0

∂E∂t

]

or

∇ ·[1μ0

(E × B)]+

12

(1μ0

∂B ·B∂t

+ ε0∂E · E∂t

)+ E ·

(J free +

∂P∂t

+1μ0

∇ ×M)= 0

(10.50)The first term on the left we interpret as the divergence of the energy flux in the Lorentzmodel:

SL =1μ0

(E ×B) (10.51)

The middle term is the rate of change of energy density stored in the fields B and E.Absent free charges or ponderable matter, these two terms state that the energy flowaway from a point in space is equal to the time rate of field energy density decrease atthat point. When matter is present, the third term on the left adds the time rate of workdue to the E-field interacting with the current density (both free and bound currents).We usually think of this third term as some kind of ‘Joule heating’ where incoming lightflux excites currents within the material and leads to a dissipative energy sink. But thisterm can also be a source of energy flux as well as a sink. If the motion of the currents isoscillatory, for example, they can be a source of radiation.

10.4.5 Energy conservation—magnetic dipole model

An analogous field divergence expression can be obtained from Equations 10.45 and10.46 resulting in

∇ · (E×H) +12

(μ0∂H ·H∂t

+ ε0∂E · E∂t

)+[E ·

(J free +

∂P∂t

)+H · ∂M

∂t

]= 0 (10.52)

Again we identify the first term on the left with the divergence of the energy flux in themagnetic dipole model and recognise the usual expression for the Poynting vector:

SMD = E ×H (10.53)


The middle term is again the rate of change of field energy density, and the third termrepresents the time rate of work done, on or by, the E-field acting on electric currentdensity (free and bound) plus time rate of work done, on or by, the H-field acting onthe magnetic current density. In free space with no sources or ponderable matter, thethird terms in Equations 10.50 and 10.52 vanish and the expressions become identical.When magnetic matter (μ �= μ0,M �= 0) is present, the two expressions for the energyflux, Equations 10.51 and 10.53, differ by a term involving the permeability and themagnetisation,

SL = SMD +1μ0

(E ×M) (10.54)

The extra term in the Lorentz version of the energy flux is sometimes called ‘hiddenenergy’ [12] and arises at the interface between two materials with different perme-abilities. We will discuss hidden energy and hidden momentum at greater length inSection 10.4.10.

10.4.6 Force law and momentum conservationin the Lorentz and magnetic dipole models

It was mentioned above in Equation 10.43 that the generalised Lorentz force law couldbe written down using ρtotal = ρfree + ρbound and J total = J free + Jbound. Carrying out therelevant substitutions and making use of vector field identities, we find that

FL = (ε0∇ · E)E +(

1μ0

∇ ×B – ε0∂E∂t

)×B (10.55)

The cross-product term on the right can be expanded so that Equation 10.55 can bewritten as

FL = (ε0∇ · E)E +(

1μ0

∇ ×B – ε0∂E∂t

)×B

= (ε0∇ · E)E +1μ0

(∇ ×B)×B –∂(ε0E ×B)

∂t+ ε0E × ∂B

∂t(10.56)

where we have used

∂ε0E ×B∂t

= ε0E × ∂B∂t

+∂ε0E∂t×B

Substituting Faraday’s law in the last term on the right in Equation 10.56 we can nowwrite it as

FL = (ε0∇ · E)E +1μ0

(∇ ×B)×B –∂(ε0E ×B)

∂t+ ε0(∇ × E)× E (10.57)


and using vector-field identity, EquationC.59, in the second and fourth terms ofEquation 10.57, we have

FL = (ε0∇ · E)E –1μ0

[12∇(B ·B) – (B ·∇)B

]–∂(ε0E ×B)

∂t– ε0

[12∇(E · E) – (E ·∇)E

](10.58)

Regrouping terms we have

FL = ε0 [(∇ · E)E + (E ·∇)E]–[ε0

12

∇(E · E)]–[1μ0

12

∇(B · B)]+

1μ0

(B·∇)B–∂(ε0E ×B)

∂t(10.59)

The first two terms on the right can be written as the divergence of the second ranktensor resulting from the direct product of E with itself:

∇ · ←→EE = (∇ · E)E + (E ·∇)E (10.60)

It is possible to get a similar divergence relation for the B field by adding 1/μ0(∇ · B)Bto Equation 10.59. The addition is permissible because ∇ · B = 0. We then can writeEquation 10.59 as

FL = ε0∇ ·←→EE +

1μ0

∇ · ←→BB –12

[ε0∇(E · E) + 1

μ0∇(B ·B)

]–∂(ε0E ×B)

∂t(10.61)

Finally, the third square-bracketed term on the right can be included in the tensor diver-gence by converting E · E and B · B to tensor form by use of the identity tensor I. Thegeneralised Lorentz force density expression is then

FL = ∇ ·[←→EE +

←→BB –

12(E · E + B ·B)↔I

]–∂(ε0E ×B)

∂t(10.62)

The last term on the right of Equation 10.62 is the time rate of change of the field mo-mentum density GL = ε0E × B. Note that GL = SL/c2. The term in brackets is called

the Maxwell stress tensor denoted by↔T . Note that the first term on the right is the di-

vergence of a tensor, not a vector. Finally, we can write the conservation of momentumcondition in analogy with the conservation expressions for charge, Equation 10.48, andfor energy, Equation 10.49:

∇ · T –∂(ε0E ×B)

∂t– FL = 0 (10.63)

The divergence of the stress tensor can be considered the time rate of momentum pass-ing through a unit of area normal to the momentum flux. Notice that the signs of thespatial and temporal derivatives are opposite (compare Equation 10.49) so that the nega-tive of the momentum flux flowing out of a closed volume (∇ · –T ) results in an decrease


in the field momentum, –∂(ε0E × B)/∂t, within the volume. We will return to discussthe physical significance of this generalised Lorentz force density expression after aninterlude on tensor analysis.

10.4.7 Digression on tensor calculus

This section sketches some essential properties of tensors, their application to elec-trodynamics, and the meaning of the notation first introduced in Equation 10.60. Forreaders already familiar with tensors and their physical significance, this section can beskipped and they can proceed directly to Section 10.4.8. Others may find the sectionworthwhile as a brief summary or review of tensor properties.

In 3-D space a second rank tensor is nothing more than a 3×3 array consisting of nineterms that transform under rotation according to certain rules. It can be generated bythe direct or outer product of two vectors. For example, the direct product of a columnvector and a row vector results in a 3× 3 matrix:⎛

⎝a1a2a3

⎞⎠× (b1 b2 b3) =

⎛⎝ab11 ab12 ab13ab21 ab22 ab23ab31 ab32 ab33

⎞⎠ (10.64)

where the first index indicates the row and the second index the column of matrix term.Not every 3 × 3 matrix, however, is a second rank tensor. Suppose that we have

two field vectors E,D in a 3-D space that are related by a linear transformation in thefollowing way:

D1 = T11E1 + T12E2 + T13E3

D2 = T21E1 + T22E2 + T23E3 (10.65)

D3 = T31E1 + T32E2 + T33E3

A second rank tensor is a linear transformation of the components of vector E intothe components of vector D that is invariant to coordinate rotation3. In matrix formEquation 10.65 is written as⎛

⎝D1

D2

D3

⎞⎠ =

⎛⎝T11 T12 T13

T21 T22 T23

T31 T32 T33

⎞⎠ ·

⎛⎝E1

E2

E3

⎞⎠ (10.66)

or more compactly,

Dj =∑k

TjkEk (10.67)

3 Here, vector E and vector D are just generic vectors, not the electric and displacements fields of Maxwell’sequations.


Under coordinate rotation, the coefficients of the vector-field components change asthe direction cosines with the condition that the absolute value (length) of the vectorremain invariant. In 3-D space a position vector r, specifying a point P with respect toan origin O, can be written as

r = x1i1 + x2i2 + x3i3 (10.68)

After a coordinate rotation the same vector can be written as

r = x′1i1′ + x′2i2

′ + x′3i3′ (10.69)

The coordinates of P after rotation are

x′j = r · i′j = x1i1 · i′j + x2i2 · i′j + x3i3 · i′j (10.70)

In order to preserve the vector length we must have

3∑j=1

x2j =3∑j=1

xj ′2 (10.71)

Now the coordinate rotation can be written in the compact notation as

x′j =3∑k=1

ajkxk (j = 1, 2, 3) (10.72)

where ajk are the terms of the 3× 3 rotation matrix. So from Equation 10.70 we see that

ajk = i′j · ik (10.73)

are the direction cosines of the rotation.Now we can determine the constraints on the coordinate rotation matrix, using

Equations 10.71 and 10.72:

3∑j=1

xj ′2 =

3∑j=1

(3∑i=1

ajixi

)(3∑k=1

ajkxk

)=

3∑i=1

3∑k=1

xixk

⎛⎝ 3∑

j=1

ajiajk

⎞⎠ (10.74)

But Equation 10.74 can only be true if

3∑j=1

ajiajk = δik δik = 1, i = k δik = 0, i �= k (10.75)


Linear coordinate transformations Equation 10.72 subject to Equation 10.75 are calledorthogonal transformations and the matrices are called orthogonal as well. Written out inmatrix notation Equation 10.75 states that if the matrix A is orthogonal,

A =

⎛⎝a11 a12 a13a21 a22 a23a31 a32 a33

⎞⎠ (10.76)

then A has the following properties:

a211 + a221 + a

231 = 1

a212 + a222 + a

232 = 1 (10.77)

a213 + a223 + a

233 = 1

but

a11a12 + a21a22 + a31a32 = 0

a11a13 + a21a23 + a31a33 = 0 (10.78)

a12a13 + a22a23 + a32a33 = 0

The orthogonality property can be easily remembered by considering the matrix ascomposed of three column vectors. The ‘dot product’ of a column with itself or witha neighbour reproduces the orthonormal rule Equation 10.75. Strictly speaking, Equa-tion 10.75 applies to the product of A with itself (by multiplying columns and summingrows as indicated by Equation 10.74), and it is evident from Equations 10.77 and 10.78that the determinant of the product matrix is unity:

det (A · A) = 1 (10.79)

but

det (A · A) = detA · detA = 1 (10.80)

and therefore

detA = ±1 (10.81)

When detA = 1, the orthogonal matrix corresponds to a true geometric rotation. WhendetA = –1 the matrix operation corresponds to an inversion followed by a rotation.

The transpose of A is given by

A′ =

⎛⎝a11 a21 a31a12 a22 a32a13 a23 a33

⎞⎠ (10.82)


and using ordinary matrix multiplication we find that

A · A′ = I (10.83)

where I is the identity matrix. Multiplying both sides of Equation 10.83 by A–1 showsthat the transpose of A is equal to its inverse:

A′ = A–1 (10.84)

Therefore, the reverse rotation from Equation 10.72 is

xk =3∑j=1

akjx′j (10.85)

the transpose of the original rotation matrix.Clearly, the orthogonal transformation must apply to vectors as well as point

coordinates. From Equation 10.73 and taking into account that i′j · ik = ik · i′j :

E ′j = E · i′j =3∑k=1

Ekik · i′j =3∑k=1

ajkEk (10.86)

In order for second rank tensors to correspond to physical entities they must also beinvariant to coordinate rotations. Start with our linear transformation Equation 10.67:

Dj =3∑k=1

TjkEk (j = 1, 2, 3) (10.87)

We seek a second rank tensor so that, after a coordinate rotation, the following relationapplies:

D′i =3∑l=1

T ′ilE′l (i = 1, 2, 3) (10.88)

where the vector components Dj ,D′i and Ek,E ′l are already related by an orthogonaltransformation. Thus, we need to find the linear transformation,

Tjk→ T ′il (10.89)


where T ′il has the property of Equation 10.88. We can relate Equation 10.87 to Equa-tion 10.88 by multiplying Equation 10.87 by aij and summing over the index j:

3∑j=1

aijDj =3∑j=1

3∑k=1

aijTjkEk (10.90)

But since D′i ,Dj and E ′l ,Ek are related simply by rotations, we can also write

D′i =3∑j=1

aijDj and Ek =3∑l=1

alkE ′l (10.91)

Note the sum on the row and not the column in the reverse operation on the right. Nowsubstitute the expression for Ek into Equation 10.90:

3∑j=1

aijDj =3∑j=1

3∑k=1

aijTjk3∑l=1

alkE′l (10.92)

D′i =3∑l=1

3∑j=1

3∑k=1

aijalkTjkE ′l (10.93)

and we identify the ‘rotated’ second rank tensor, T ′il , as

T ′il =3∑j=1

3∑k=1

aijalkTjk (i, l = 1, 2, 3) (10.94)

The second rank tensor Tjk that transforms as Equation 10.94 is invariant to coordinaterotation and is admissible to represent physical quantities.

We can easily show that the scalar product of two vectors is invariant to coordinatetransformation as well by invoking Equations 10.75 and 10.85:

D · E =3∑k=1

DkEk =3∑k=1

⎛⎝ 3∑

j=1

ajkD′j

⎞⎠( 3∑

i=1

aikE ′i

)=

3∑k=1

⎛⎝ 3∑i,j=1

ajkaikD′jE′i

⎞⎠ =

3∑k=1

D′kE′k

(10.95)

The gradient of a scalar function ϕ is invariant to orthogonal transformations:

∇ϕ =3∑k=1

∂ϕ

∂xkik (10.96)


Consider one of the terms in the sum and set

Di =∂ϕ

∂xi(10.97)

and take the derivative of xk with respect to x′i in Equation 10.85:

∂xk∂x′i

= aik (10.98)

Then,

D′i =∂ϕ

∂x′i=

3∑k=1

∂ϕ

∂xk

∂xk∂x′i

=3∑k=1

aikDk (10.99)

and

3∑i=1

D′ii′i = ∇′ϕ =

3∑j=1

3∑k=1

aikDiij =3∑k=1

aik∇ϕ (10.100)

This last expression shows that the gradient of scalar function transforms as a vector.In addition to the scalar product and gradient operations we can show that the di-

vergence operation also is preserved under orthogonal transformations. Consider twovectors D and E and suppose that

Ei =∂Di

∂xi(10.101)

Then

E ′i =∂D′i∂x′i

=3∑k=1

aik∂D′i∂xk

(10.102)

where we have used from Equation 10.98:

∂

∂x′i=

3∑k=1

∂

∂xk· ∂xk∂x′i

=3∑k=1

aik∂

∂xk(10.103)

Now making a substitution from Equation 10.86 into the right-hand side of Equa-tion 10.102 we have

E ′i =∂D′i∂x′i

=3∑k=1

aik∂D′i∂xk

=3∑k=1

aik3∑j=1

aij∂Dj

∂xk=

3∑k=1

3∑j=1

aikaij∂Dj

∂xk(10.104)


Finally, summing over the index i in this last expression and using the Kronicker δproperty of Equation 10.75 we have:

∇′ ·D′ =3∑i=1

∂D′i∂xi

=3∑j=1

∂Dj

∂xj= ∇ ·D (10.105)

So we see that the divergence of a vector is invariant to orthogonal transformations.The divergence of a tensor is defined as

(∇ · ↔T

)j=

3∑k=1

∂Tjk∂xk

= Dj (j = 1, 2, 3) (10.106)

We can show that the divergence operation on a second rank tensor results in a vec-tor (first rank tensor), and that therefore, since we have shown above that a vector isinvariant to orthogonal coordinate transformation, the same can be said of the tensor di-vergence. Equation 10.106 shows that the divergence operation on the tensor is similarto the operation on a vector except the derivative is taken on each row and summed onthe columns. The result is three terms that constitute the vector components Dj . In orderto show the invariance of the divergence with respect to coordinate rotations we followthe same steps as for the vector divergence, applying the procedure to each row:

D′i =3∑l=1

∂T ′il∂x′l

=3∑k=1

3∑j=1

alkaijalk∂Tjk∂xl

=3∑k=1

3∑j=1

aijalkalk∂Tjk∂xk

=3∑j=1

aij

(3∑k=1

∂Tjk∂xk

)=

3∑j=1

aijDj (10.107)

In the middle term above we have again taken advantage of the Kronicker δ property.The expression Equation 10.107 shows that each component of the vector resulting fromthe divergence operation on the tensor transforms as a coordinate rotation. Therefore,

∇′ ·↔T ′ = ∇ · ↔T (10.108)

Finally, the origin of the double-arrow notation arises from considering the secondrank tensor a dyadic. A dyadic is similar to a vector but is flanked by unit vectors on theright and the left. Thus, writing the tensor as a dyadic,

↔T =

3∑i=1

3∑j=1

εiTijεj (10.109)


where εi, εj are unit vectors along the 3-D orthogonal directions. An individual elementof the tensor matrix is found by performing the scalar product of the unit vectors on theleft and right:

Tij = εi ·↔T · εj (10.110)

It is easily seen that taking the divergence of a dyadic tensor results in a vector:

∇ · ↔T =3∑i=1

∂

∂xiεi ·

3∑i=1

εiTij3∑j=1

εj =3∑j=1

∂Tij∂xi

εj (i = 1, 2, 3) (10.111)

The dyadic notation is reminiscent of Dirac bra, ket notation in quantum mechanicswhere the basis space is represented by kets, the ‘dual’ space represented by bras, andthe operators are flanked on the left and right by kets and bras, respectively.

10.4.8 Return to Lorentz force law

Now armed with a better understanding of second rank tensors and dyadic notationwe can reconsider the physical significance of Equation 10.62, reproduced here forconvenience:

FL = ∇ ·[←→EE +

←→BB –

12(E · E + B ·B)↔I

]–∂(ε0E ×B)

∂t

We imagine a body in space subject to external forces and enclosed by a volume. Weobtain the mechanical total force on the body by integrating Equation 10.62 over thevolume:

FLtotal =∫FL dV =

∫ ↔T · da –

∫∂(ε0E ×B)

∂tdV (10.112)

where da is the surface differential of the enclosing volume, and we have used the tensorversion of the divergence theorem:∫

∇ · ↔T dV =∫ ↔

T · da

Note that the Lorentz energy flux Equation 10.51 and the Lorentz field momentumε0 E × B are related by

1c2

1μ0

(E × B) = ε0 E ×B (10.113)

Equation 10.112 states that the total mechanical force within the volume is equal to thedifference between the force applied to the enclosing surface and the time rate of change


of field momentum within the volume. Or, alternatively, by bringing the last term onthe right over to the left, we can write an expression for the conservation of momentumstating that the sum of the rate of change of mechanical momentum (FLtotal) and the rateof change of electromagnetic field momentum is equal to the total stresses (compressive,tensile, and shear) at the enclosing surface:

FLtotal +∫∂(ε0E ×B)

∂tdV =

∫ ↔T · da (10.114)

In general, the enclosed volume does not have to be commensurate with the body, but itmakes the physical interpretation easier if we assume that the enclosing volume is just aninfinitesimal distance outside the body. Then the term on the right of Equation 10.114describes the integral of the stress tensor over the positive outward normal body surfaceelement da. The forces on the body are of two kinds: the force components alignedalong the surface element components normal to the coordinate axes (diagonal tensorterms) and force components tangent to these surface elements (off-diagonal tensorterms). The former constitute components of compression or tension (depending onthe sign) while the latter are shear forces. In static equilibrium the stress tensor mustbe symmetric, Tij = Tji. However, we do not really have to imagine a ponderable bodyinside the enclosing volume on which tensile and shear forces bear. Even if the volumeonly encloses the vacuum, Equation 10.114 is valid. In any case, we interpret a positiverate of increase in the field momentum (second term on the left-hand side) as due to theflow of momentum across the surface in the negative direction to the surface normal. AMaxwell stress tensor element –Tij can be considered the momentum per unit time inthe i direction crossing the surface element da whose normal is oriented along the j-axis.

10.4.9 Einstein–Laub force law

The Lorentz force law is almost universally applied to connect Newtonian mechanics toMaxwellian electromagnetics. However, it is not the only force law extant. Several otherstress tensors have been proposed by such notables as Minkowski, Abraham, Peierls, andL. J. Chu. A recent review [13] tabulates five principal proposals. Among these, in thefirst decade of the twentieth century, A. Einstein and J. Laub [14] suggested a force lawthat, from the perspective of physical interpretation, may have certain advantages overthe conventional Lorentz law [12]:

FEL = ρfreeE+J free×μ0H+(P·∇)E+(∂P∂t× μ0H

)+(M·∇)H–

(∂M∂t× ε0E

)(10.115)

Following the procedure analogous to the development of the generalised Lorentz forceexpression (Equations 10.55–10.62) we can cast Equation 10.115 into a similar forminvolving only vector fields and stress tensors:


FEL = ∇ ·[←→DE –

←→BH –

12(ε0E · E + μ0H ·H)

↔I]–∂(E ×H/c2

)∂t

(10.116)

where the expression in square brackets is the Einstein–Laub stress tensor:

T EL = ∇ ·[←→DE –

←→BH –

12(ε0E · E + μ0H ·H)

↔I]

(10.117)

Integrating over the volume on both sides of Equation 10.116, applying the divergencetheorem and rearranging terms, results in a conservation of momentum expressionsimilar to Equation 10.114, and we obtain

FELtotal +∫∂(E ×H/c2

)∂t

dV =∫ ↔

T EL · da (10.118)

Note that the rate of change of field momentum is now expressed in terms of the crossproduct of E and H rather than E and B. The time rate of change of field momentumGEL is related to the Abraham energy flux or the familiar Poynting vector S:

GEL =E ×Hc2

=Sc2

(10.119)

which we have labelled SMD in Equation 10.53 to indicate that it arises naturally fromthe magnetic dipole model for ponderable matter with magnetisation (Section 10.4.3).In fact, the Einstein–Laub energy flux and force expressions are ‘natural’ to the Max-well equations, Equations 10.44–10.47, since they all use E and H. In contrast, theLorentz energy flux and force expressions are appropriate to the Maxwell equations,Equations 10.36–10.39, since they all use E and B.

It is important to bear in mind that although the Lorentz and Einstein–Laub lawsare formulated from the same elements (charges, currents, and fields) as the Maxwellequations, they are not deduced or derived from them. An equation relating electromag-netic fields to mechanical forces is in fact an additional postulate to the four equationsof Maxwell. Forces entail the time derivative of momenta, and we have already seenthat the Abraham and Minkowski versions of field momentum still have their ardentproponents over a century after their introduction. In addition to the Lorentz andEinstein–Laub force laws, therefore, we can expect to, and do, encounter others. Theyhave been discussed in recent reviews, [13] [15], [16]. It turns out that the global prin-cipals of conservation of energy and momentum enforce the same result for the total,integrated energy, momentum, and force expressions over a given volume, but the forcedensities (and therefore the momentum density distributions) within a given volume arenot identical. Experiments of light forces acting on deformable bodies, such as liquidsor soft solids, should in principle, enable comparison of the different predicted forcedistributions, but so far designing the definitive experiment has proved elusive.


10.4.10 Hidden energy and hidden momentum

The terms ‘hidden energy’ and ‘hidden momentum’ [17] do not refer to a single, well-defined phenomenon, but have been invoked from time to time to account for apparentlymissing or unobservable quantities needed to preserve the energy and momentum con-servation laws. Hidden energy arises from the choice of expression for energy flux(Poynting vector) in electromagnetic fields, and hidden momentum is generally asso-ciated with a body whose centre-of-mass (or centre-of-energy) is at rest but has aninternal structure that somehow possesses momentum. Hidden quantities are related tothe long-running dispute over Abraham and Minkowski momentum. The most refrac-tory controversies arise from the different, legitimate ways to parse and interpret certainterms in Maxwell’s equations, as well as the consequent choice of energy flux expres-sion and force law. A thorough discussion, analysis, and evaluation of the various pointsof view is beyond the scope of this book and has been extensively examined elsewhere[9], [11]–[13], [15]–[21]. We will content ourselves, here, with an illustrative example(closely analysed and discussed at greater length by [12]) of the difficulties encounteredin the presence of magnetic media (μ �= μ0,M �= 0).

Imagine again a quasi-monochromatic light pulse of time length T , propagating infree space and then impinging on a material slab with permittivity ε and permeabilityμ so that μ = ε but both are greater than ε0,μ0. The vacuum and the material areimpedance-matched so there will be no reflection at the interface. Figure 10.6 depicts thesituation. The E,H field components parallel to the slab surface are continuous throughthe interface but the B-field is not. Therefore, if we use the Lorentz form of the Poyntingvector, SL = μ–1

0 (E×B), the power flux will also be discontinuous at the interface. In fact,there will be a sudden increase in power flux as the light pulse propagates from free spaceinto the material slab due to the μ–1

0 (E×M) term of Equation 10.54. In order to maintainenergy conservation there must be an amount of ‘hidden energy’ somewhere to accountfor the discrepancy. In contrast, the Poynting vector associated with the magnetic dipole

light pulsesource

Ex

By

kz

ε0, μ0 ε, μ ε = μJsFigure 10.6 A quasi-monochromatic lightpulse of duration T propagates from vacuuminto a material slab with permittivity andpermeability ε and μ, respectively, and withε = μ. The media are impedance-matched,and there are no reflections at the surface.A bound surface current Js appears at theinterface while the pulse enters the slab. Thepulse edges should be considered abrupt withHeaviside-like changes in the field amp-litudes and with near-constant amplitudeduring the pulse length T.


model of magnetisation (Section 10.4.3) or the Einstein–Laub force law, S = E × H,is continuous across the boundary and maintains energy conservation without recourseto hidden sources. In the case of the Lorentz form for the Poynting vector the ‘missingenergy’ is supplied by a bound current induced at the slab surface as the light pulsepasses into it. The term responsible is E · (μ–1

0 ∇ ×M) in Equation 10.50. As the lightpulse moves into the material, the scalar product of the E-field and the surface currentsupply the energy needed to maintain the energy balance at the interface. When thepulse exits the slab, this energy is returned to the material via the same surface currentinteraction. In the magnetic dipole model, the magnetisation contributes to the energyflux through the H · ∂M/∂t term of Equation 10.52, but since the magnetisation is time-independent, the contribution is null, and the induced surface current does not couple tothe pulse energy. Thus, we see that the conventional S = E×H provides a much simplerroute to energy conservation. The origin of material magnetisation in the Lorentz modelis the amperian current loop. One imagines a tiny circulating disc with charge current iand fixed diameter r giving rise to a magnetic moment m = μ0iA, where A is the area ofthe circulating disk plane. The macroscopic magnetisation M = N m, where N is thenumber density of the individual magnetic moments. But even if this model is admitted,it is difficult to understand how to interpret the flow of energy into or out of such anamperian current loop. Adopting S = E×H avoids hidden energy and the current loopmodel, but the price to be paid is a force law that is not a straightforward generalisationof the microscopic Lorentz force law.

Further close analysis [12] reveals that, under the Lorentz model, the slab is subjectto an impulsive force of equal amplitude but opposite sign at the leading and trailingedges of the light pulse. In addition, as the pulse enters the slab, the vector productof the surface bound current and the B-field produces a force that exactly cancels theimpulsive force of the leading edge. Altogether no net force is applied to the slab as thepulse passes through it and therefore no momentum is transferred to the slab. The initialmomentum of the light pulse prior to incidence on the slab is 1/2ε0E2

0 . But according tothe Balazs thought experiment (Section 10.3) the momentum carried by the pulse insidethe slab must be reduced by a factor of

√με. Therefore, the Lorentz bound-current

model results in a momentum deficit of 1/2ε0E20(1 – 1/

√με), the ‘missing momentum’.

Using the alternative force law of Einstein–Laub, the bound surface currents Js do notcouple to the light, and therefore, the cancelling force on the slab due to Js × B is notpresent. The impulsive force due to the pulse leading edge is, however, still operative,and so during the time of the pulse entry, the slab does receive a forward kick. Themomentum acquired by the slab is 1/4ε0[(ε + μ – 2)/μ]E2

0T z, where T is the temporalpulse length. Remembering that we have set μ = ε, the mechanical ‘kick’ momentumacquired is therefore 1/2ε0(1 – 1/

√εμ)E2

0T z. The sum of the slab mechanical and lightpulse momentum is therefore:

12ε0

(1 –

1√εμ

)E20T z +

12ε0

1√εμE20T z =

12ε0E2

0T z (10.120)


the momentum of the light pulse prior to impinging on the slab. Therefore, the Einstein–Laub force law results in a full accounting of momentum without the need to resortto ‘hidden’ terms. Shockley and James [22] first drew attention to missing momentumin magnetic media and identified it with amperian current loops responsible for themagnetisation in the Lorentz model.

10.4.11 Key experiments

In this section we will describe two experiments that have attempted to provide decisiveevidence to resolve the Abraham–Minkowski controversy (Section 10.3.1).

10.4.11.1 Radiation pressure on a mirror immersed in a dielectric fluid

The first experiment involves measuring the radiation pressure on a mirror immersed ina variety of dielectric fluids with different indices of refraction. The experiment was firstcarried out by Jones and Richards in 1954 [23] and later repeated with much improvedtechnology in 1978 [5]. The idea is to measure the radiation pressure on the mirroras a function of the index of refraction so as to determine if the force, and thereforethe rate of change of momentum of the light reflecting from the mirror, varies directlywith the refractive index (Minkowski) or inversely (Abraham). Figure 10.7 shows insome detail the torsion balance setup to measure the force on the immersed mirror.A laser beam impinges alternately on the left and right sides of the radiation pressuremirror, which itself is surrounded by a dielectric organic liquid of accurately knownrefractive index at the wavelength of the helium-neon laser. The ratio of the radiation

torsion head

insulating collar

optical levermirror and coil

radiation pressuremirror

optical leverwindow 8 mmdiameter

‘angled’windows24 mm high9 mm wide

liquid outlet

laser beamstrip50 × 5 μm40 mm long

balance mass

container intwo halvesclamped withan indium seal

liquid inlet &level gauge

15-turn coil

Helmholtzcoils 2 × 50 turns66 × 40 mm30.5 mm apart

coil spacer0.152 mm

optical levermirror2 × 5 × 0.2 mm

radiation pressuremirror5 × 2.5 × 0.1 mmmultilayer coated

copper wire0.04 mmdiameter50 mm long

web to reduceliquid spacetension spring

frame to gointo container

cross sectionthroughr.p. mirror

5 cm

10 cm

Figure 10.7 Detail of the torsion balance: (left) torsion balance suspension frame, (centre) containerinto which the frame is introduced, (right) detail of the radiation pressure mirror and suspension.Figure from ref. [5] and used with permission.


pressure on the deflection mirror in the liquid to that in air was measured for sevendifferent organic liquids, and the ratios were compared to the phase refractive indices.The ratios agreed to within a ‘mean discrepancy, averaged over the seven liquids, of–(0.003 ± 0.053)%’ [5], which would appear to confirm that the light propagates in thedielectric with the Minkowski momentum. Or does it? Analysis and interpretation ofthese results has led to widely varying conclusions. Mansuripur [24] has pointed outthat while the experimental results are certainly convincing, the force on the submergedmirror is a sensitive function of the phase shift between the incident and reflected light. Amirror with a Fresnel reflection coefficient near unity in which the E-field changes phaseby π at the surface (the usual case) will indeed result in a Minkowski-like momentum;but the complex reflection coefficient is expressed by r = |r|eiϕ , and if ϕ = 0, then themirror would experience a force consistent with the Abraham momentum. In fact, themeasured radiation pressure would vary continuously between the two limiting cases as afunction of the phase angle of the mirror. The conclusion is that the experiment does notmeasure the intrinsic momentum of the light wave in the dielectric fluid independentlyof the phase of the Fresnel reflection coefficient. The expression for force density on themirror is given by [24]:

〈Fz〉 =[1 +

(n20 – 1

)sin2

(ϕ2

)]ε0E2

0 (10.121)

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

ϕ (°)

Str

ess

(N/m

2 )

EMMaterialTotal

PEC (ϕ = 180)PMC (ϕ = 0)

Figure 10.8 Stress (Nm–2) vs. mirror phase angle φ. Triangles are the electromagnetic stress,open circles the material stress of the dielectric liquid, and the solid line is the sum = 2nε0E2

0 ,twice the Minkowski momentum. PMC=perfect magnetic conductor, PEC=perfect electricconductor. The index of refraction n = 2. Figure taken from [13] and used with permission.


where Fz is the force density on the mirror averaged over an optical cycle and inte-grated over the mirror penetration depth, n0 the refractive index of the dielectric fluid, ϕthe Fresnel phase, and E0 the light wave E-field amplitude.

But that interpretation and conclusion have not gone unchallenged. Kemp andGrzegorczyk [25], and later Kemp [13], agree that the Mansuripur calculation for theelectromagnetic force is correct but that it is not the only force bearing on the mirror.Calculating the forces from the Minkowski stress tensor, they claim the presence of anadditional force at the mirror-dielectric interface, due to the dielectric itself, that effect-ively restores the Minkowski result for all mirror phases. Their expression [25] for theforce density on the mirror is

Ftotal = Fl + Fm = 2n0ε0E20 (10.122)

where Fl ,Fm are the forces due to the dielectric liquid and the EM field, respectively.Figure 10.8 shows the calculation of the two forces as a function of mirror phase andtheir sum, seen to be independent of phase. Mansuripur [26] disputes the validity of thematerial force term which is dependent on the choice of stress tensor. The two calcula-tions do not use the same approach to obtain the net force. Mansuripur calculates theforce on the mirror directly from the Lorentz force law while Kemp calculates it from theMinkowski stress tensor. In principle both approaches are valid. It appears that the dis-agreement could be finally resolved by an experiment measuring the radiation pressurein a series of mirrors with varying phases and little loss. The technology for fabricatingsuch mirror coatings is readily available.

10.4.11.2 Atom recoil in a Bose–Einstein condensate

The immersed-mirror experiments involve light propagating through condensed, con-tinuous media. At the other extreme of matter density, light of wavelength λ andfrequency ν in the optical range scatters from individual atoms. Since the light inter-acts with quantised atomic internal states, one usually considers the light as photonswith energy hω and momentum hk, where ω is the angular light frequency, ω = 2π /νand k = 2π /λ, even though the light field itself is classical. If a contained ensemble ofboson atoms is cold enough and reaches a critical density in the containing volume, itwill condense into a collective quantum state called a Bose–Einstein condensate (BEC).Although the BEC can be characterised by a wave function with a unique phase, it isstill a highly dilute gas. Typical densities are � 1014 atoms cm3 implying an average dis-tance between the atoms of the condensate � 200 nm. Considering that the atomic Bohrdiameter � 10–10 m, we see that the condensate consists mostly of empty space.

Here we consider an experiment in which two separate light pulses interacting witha BEC probe the atom recoil momentum by atom interferometry [6]. The schematic ofthe experiment is shown in Figure 10.9. The experiment consists of a two-pulse atominterferometer. The first pulse sets up a standing wave in the 87Rb BEC. The laser istuned near (above and below but not directly on) the 52S1/2, F = 1 ↔ 52S3/2, F = 1transition at λ = 785nm. The pulse duration is long enough such that the laser linewidth


Release from TrapImage in TOF

600 μs

5 μs 5 μs

B

E

τ

+2nhk

0nhk

–2nhk

Figure 10.9 Schematic of the two-pulse,Ramsey-type interferometer (Kapitza-Diracinterferometer). The first laser pulse on theleft sets up a standing wave and scatters asmall fraction of the 87Rb BEC atoms into| ± 2nhk > momentum states. The phasesof the ground- and excited-state wave func-tions evolve at different rates. After the variableinterval τ a second pulse projects the excitedstates back onto the ground state. The laserpropagates perpendicular to the plane of thefigure with the E-field polarised along the longaxis of the BEC. A static magnetic field B,also aligned along the same axis, provides anaxis of quantisation. The probability of find-ing the atoms in the | ± 2nhnk > momentumstates is measured by a time-of-flight (TOF)imaging technique shown on the right. Figurereproduced from [6] with permission.

is much smaller than the spontaneous emission linewidth of the atom, and the laser fre-quency offset from the atom resonance centre frequency is always much greater than thenatural linewidth. In the presence of the standing wave, a small fraction of the atoms inthe |0nhk > ground momentum state scatter into | ± 2nhk > excited momentum states,experiencing a momentum ‘kick’ and gaining a recoil energy. In free space an isolatedatom absorbing a photon gains a recoil energy of ER = hωR = h2k2/2m, where k = 2π /λand m is the atomic mass. The BEC, although a very dilute gas, nevertheless presents anon-negligible index of refraction n, and the essential idea of the experiment is to meas-ure how the recoil momentum depends on n. The standing-wave light field produces amomentum grating that scatters the atoms of the BEC into a mixed momentum-statewave function,

|ψ(τ) >= J0|0nhk > + J1| ± 2nhk > e–ihωRτ (10.123)

where J20 is the population of atoms in the |0hk > ground state and J21 is the population ofatoms in the |±2nhk >momentum states. The coefficients J0, J1 are zero- and first-orderBessel functions. In fact, there are higher order terms corresponding to higher momen-tum states but their probabilities are negligible. The phase of the | ± 2nhk > evolvesin time with the recoil energy as indicated in the second term on the right of Equa-tion 10.123. The second light pulse arriving at time τ projects the mixed momentumstate back onto the ground state with a probability modulated by this phase evolution.The expression for the modulation is

P0 = J40 + 4[J20 J

21 + J

41

]cos(4n2ωRτ) (10.124)

assuming that the n dependence of the recoil momentum is 2nhk, and therefore,ER = 4n2hωR. The measured points together with a fitted analytic curve of the form of


0.90

0.80

0.70

600400200τ (μs)

0

(b)

10 μs 20 μs 30 μs 40 μs 50 μs(a)

Fra

ctio

n in

0nh

k

Figure 10.10 Solid points: measured fraction of the atompopulation in the |0nhk > state as a function of τ . Solid line:Functional form of Equation 10.124 with damping fitted totake into account time dependence of the wave function overlap.Figure reproduced from [6] with permission.

Equation 10.124 for a fixed laser detuning is shown in Figure 10.10. The fit is seen to bevery good for the assumed form of the n dependence of the recoil momentum, |2nhk >.Furthermore, because the BEC ‘material’ exhibits dispersion around the atomic transi-tion resonance line, the refractive index will have a pronounced frequency dependencein this spectral region. A series of measurements with the laser frequency set to sev-eral ‘detunings’,� = ω – ω0 around the resonance frequency ω0, confirm that the recoilmomentum is directly proportional to n (Minkowski form).

10.4.12 Interpretation of the key experiments

But does this result mean that the ‘photons’ in the BEC carry momentum nhk? Withthe case of the immersed-mirror experiment in mind, Mansuripur and Zakharian [27]calculated the energy flux and Lorentz force carried into a generic material slab with re-fractive index n2 + iκ2 from a dielectric medium of index n1 by a plane wave propagatingalong z . They obtain an expression for the momentum transferred,


g2 =2n1

(1 + n22 + κ

22

)(n1 + n2)2 + κ22

hkzz (10.125)

For the highly reflective, lossless mirror case, κ2 is pure imaginary and its absolutevalue is much greater than n1 or n2. Application of Equation 10.125 to this case yieldsg2 � 2n1hkzz, in agreement with the previous discussion of the Fresnel reflection co-efficient phase [24]. According to Reference [27] in the Bibliography in Section 10.7[27], the apparent Minkowski result is a consequence of the high impedance mismatchbetween the two media, the liquid dielectric n1 and the mirror n2 + iκ2, rather than theintrinsic momentum carried by the light in the dielectric fluid. In the case of the BEC,n1 = 1, n2 ≈ 1.2 (see Figure 3 and Equation 4 of Reference [6] in the Bibliography inSection 10.7), and κ2 ≈ 0. Taking the refractive index of the BEC as n2 � n1 + α (withα << n1), the apparent light momentum in the BEC, according to Equation 10.125,appears as

g2 =2n1

[1 + (n1 + α)2

](2n1 + α)2

hkzz � 12

[1 + n21n1

+ 2α]hkzz (10.126)

then

limn1→1

g2 → (1 + α)hkzz � n2hkzz (10.127)

Although both experiments yield the Minkowski form in the measurements, thereis an important difference between the immersed-mirror experiment and the BECexperiment. The quantum-gas measurement senses the recoil energy (via n2ωR in Equa-tion 10.124), not the force, and therefore, is directly sensitive to the wavelength of lightin the BEC. The data fit ER = 4n22hωR, not ER = 4n21hωR, and provides a fundamentallydifferent measure of the light momentum in the BEC than the force measurement onthe mirror immersed in the liquid dielectric. Although it might appear that Minkowskiwins the day, at least in this interferometric measurement, even here appearances can bedeceiving. In addition to the impedance arguments of Mansuripur, the early study byGordon [1] shows that even in a highly dilute, weakly polarisable classical gas, the trueelectromagnetic momentum is still the Abraham term even though in force calculationsit is the Minkowski form that appears. Loudon [28] reached similar conclusions from ageneral quantum treatment of light forces on dielectrics (including lossy homogenousmaterials). Mansuripur [29], from a purely classical treatment, also found that the mo-mentum within dielectrics consists of a mechanical momentum term with the Minkowskiform and an electromagnetic term of the Abraham form. All three authors calculate themomentum density and total momentum starting from the Lorentz force expression,avoiding the stress tensor formalism altogether. There is general agreement among theseauthors that experiments measuring force do not directly probe the electromagnetic mo-mentum content contained within a ponderable medium and transferred to it from aninitial light field. Therefore, it is not permissible to infer directly the form of the total


momentum content (electromagnetic plus mechanical) from radiation pressure experi-ments. Although the Minkowski form persistently appears in the results of very carefulmeasurements, there is no gainsaying the validity of the Balazs thought experiment orthe consistent conclusions from classical and quantum analysis, all of which point to theAbraham form for field momentum in dielectrics. The nagging question still remains:is there any resolution to this seemingly intractable conundrum of which momentum,Minkowski or Abraham, is ‘correct’?

10.4.13 Kinetic and canonical momentum

Barnett and Loudon have proposed [19] that in fact there are two distinct momenta as-sociated with light: the kinetic momentum and the canonical momentum. They identifythe kinetic momentum as the Newtonian momentum equal to the product of the massand velocity of a point particle. The canonical momentum is associated with the La-grange formulation of classical mechanics in which the position and momentum are saidto be ‘canonical’ variables. They are related by

pi =∂L∂ qj

i, jth degrees of freedom (10.128)

where L = T – V is the Lagrangian of a system, pi is the momentum, and qi the gen-eralised coordinate. The kinetic energy T and potential energy V constitute the twoterms of L. At the microscopic level of a point particle with mass m and charge q theLagrangian can be written more explicitly as

L =12mv2 + qv · A – qV (10.129)

where v is the particle velocity and A is the vector potential. From Equation 10.128, thecanonical momentum is then

p = mv + qA (10.130)

The canonical variables can be used to form a function called the ‘Lagrange bracket’which possesses special invariance attributes in classical mechanics. The fundamentalLagrange brackets [30] are given by

{qi, qj} = 0

{pi, pj} = 0

{qi, pj} =∑k

(∂qk∂qi

∂pk∂pj

–∂qk∂pj

∂pk∂qi

)(10.131)


The second term on the right vanishes because p, q are independent coordinates.Therefore,

{qi , pj} =∑k

δjkδki = δij (10.132)

The Lagrange bracket is related to the more familiar Poisson bracket by

n∑i

{pl , qi}[pl , pj] +n∑i

{ql , qi}[ql , pj] = 0 (10.133)

From which it can be shown that the canonical variables p, q obey the Poisson bracketexpression

[qi, pj] = δij (10.134)

which in quantum mechanics goes over into the familiar commutation relation

[x, px] = ih (10.135)

Of course in quantummechanics px is a component of the momentum operator p = ih∇,and this form of the momentum operator was intended to correspond to the de Brogliehypothesis on matter waves, p → h/λ. Therefore, in quantum mechanics one may saythat the canonical momentum operator is associated with the wavelength of the matterwave. Reasoning by analogy, Barnett and Loudon [19] suggest that since the Minkowskimomentum appears in experiments involving diffraction and interference (the BECexperiment of Section 10.4.11, for example), where the wavelength of light is the de-termining property, it might be fruitful to associate the Minkowski momentum to thecanonical momentum. In contrast, as illustrated by the Balazs thought experiment, wherethe light simply passes from one refractive index to another ‘kinematically’ (withoutreflection or absorption), the Abraham momentum might be the appropriate choice.

10.4.14 Illustrative example—light force on a point dipole

For solid, polarisable dielectric objects with no free charges or currents, the lightforce expressions arising from the Lorentz force law can be written in two equivalentways [31, 32],

Fc = – (∇ · P)E +(∂P∂t

)×B (10.136)

Fd = (P · ∇)E +(∂P∂t

)×B (10.137)


where Fc indicates the force derived from bound charges and currents, and Fd assumesthe material is composed of point dipoles4[1]. For most practical calculations of experi-mental interest, the dipole form, Equation 10.137, is the one of choice. In Chapter 11we will discuss in some detail, from a semiclassical point of view, light forces bearingon a two-level atom (see especially Section 11.4), in which the two levels are coupled bya transition dipole matrix element. As a precursor to that chapter we summarise herethe forces and momenta acquired by a point dipole as a pulse of light passes through it.Considered optically, the electric point dipole stands in for the most primitive of atoms.The discussion is based on an important article by Hinds and Barnett [33] that illus-trates how the Abraham and Minkowski momenta may be associated with qualitativelydifferent aspects of the light–dipole interaction.

To summarise the preceding discussion (Sections 10.4.1–10.4.6, and 10.4.8), theAbraham and Minkowski expressions for the momentum density of light are,

GA =1c2E ×H and GM = D×B (10.138)

In free space these expressions are equivalent, but when light travels in a medium ofrefractive index n, the momentum of the light takes on two different expressions. Inte-grating over a defined volume and normalising the light energy to the photon, hω, theAbraham and Minkowski momenta become

pA =p0

nand pM = p0n (10.139)

where p0 is the free-space momentum of the light. The question we wish to answer is:what is the relation between pA and pM and the kinetic and canonical momenta in thecase of light interacting with a point dipole? We can begin to answer this question byciting an important relation between the kinetic and canonical momenta [19] when anoptical field interacts with a dipole d,

pkin = pcan + d ×B (10.140)

and integrating over the volume of a medium consisting of dipoles d,

pkin – pcan =∫

(gM – gA) dV = d ×B (10.141)

or

pcan +∫gM dV = pkin +

∫gA dV (10.142)

4 The form of the first term in Equation 10.137 is always used in atomic physics and arises from firstconsidering the dipole-E-field interaction energy, E = –d · E, followed by taking the negative of the spatialderivative to get the force while taking into account that the spatial extent of the optical wave is always muchgreater than that of the atomic dipole. See Reference 7 in Further reading, Section 10.6.


The canonical momentum as introduced in Section 10.4.13, is essentially a quantumoperator, not a classical entity, the form of which depends on the gauge of the cor-responding vector field. Equations 10.140–10.142 are valid for the Coulomb gauge.According to Barnett [34] the canonical momentum can also be cast as a classical en-tity, and the choice of Coulomb gauge rather than some other arbitrary gauge does notmaterially effect the results. The choices here are for convenience and clarity.

Starting from Equation 10.137, the force component Fi acting on a point dipole d bya quasi-monochromatic optical pulse E = E0(z)e cos(ωt – kz), propagating along z andwith unit vector e in the x – y plane, can be expressed in the form of a Lorentz force as,

Fi = (d ·∇)Ei +(d ×B)i (10.143)

or, invoking the identity,

∂

∂t(d ×B)i = (d ×B)i + (d × B)i (10.144)

and Faraday’s law,

B = –∇ × E (10.145)

we can, after using a vector triple product identity, write

Fi = d · ∂∂xi

E +∂

∂t(d ×B)i (10.146)

Now substitute the E-field of the pulse, E = E0(z)e cos(ωt – kz) into Equation 10.146.The result is

Fz = F1z + F2z = d · e[∂E0(z)∂z

cos(ωt – kz) + E0(z)k sin(ωt – kz)]+∂

∂t(d ×B)i

(10.147)We will see in Chapter 11 that the light force on an atom appears in a form similar to

the first term in square brackets in Equation 10.147. The second term does not appearin the conventional force expressions for atom optical manipulation, but we can call itthe Lorentz term, because d is analogous to a bound polarisation current and d × B isa Lorentz force term. Up to this point we have been able to consider the fields and thedipole as classical quantities. The development of Hinds and Bartlett [33] now proceedssemiclassically by quantising the dipole but keeping the fields classical. We consider thepoint dipole a quantum object with two internal states: one lower (1) and one upper (2).The force Fz must now be calculated by taking matrix elements according to conven-tional quantum mechanics as described in detail in Chapter 11, Section 11.4. Here, wesimply write the result,


〈F1z〉 = 〈d · e〉[∂E0(z)∂z

cos(ωt – kz) + E0(z)k sin(ωt – kz)]

(10.148)

where 〈d · e〉 = d12 is the transition matrix element of the dipole between states (1) and(2). Around the dipole transition frequency ω0, Section 11.5.3 shows how the E-field ofthe incoming pulse drives the dipole as a forced oscillator according to

〈d · e〉 = 2d12

{[δ(ω) 12�

δ2 + γ 2 + 12�

2

]cos(ωt – kz) –

[γ 1

2�

δ2 + γ 2 + 12�

2

]sin(ωt – kz)

}

(10.149)

where δ = ω – ω0 is the detuning, γ is the generic loss rate from the upper state5, andh� = –d12E0 defines the Rabi frequency6 �. The first term on the right is the ‘disper-sive’ term that goes to zero at zero detuning and is antisymmetric on either side of ω0,while the second term is the ‘absorptive’ term that peaks on resonance. The dispersiveterm is in phase with the driving field while the absorptive term is in quadrature. Thesusceptibilities χ ′ and χ ′′ show the same behaviour and are plotted as a function of fre-quency in Figures 11.2 and 11.3. Substitution of Equation 10.149 into Equation 10.148and averaging over an optical cycle results in a two-term expression for �F1z,

�F1z = d12

[δ(ω) 12�

δ2 + γ 2 + 12�

2

]∂E0(z)∂z

– d12

[γ 1

2�

δ2 + γ 2 + 12�

2

]kE0(z) (10.150)

The dispersive term, proportional to the E-field amplitude gradient along z, is called the‘dipole-gradient force’, and the absorptive term is called the ‘radiation-pressure force’.Now we concentrate on the dipole-gradient force term and integrate it over the timerequired for the light pulse to fully overlap the dipole. The result will be the momentumdelivered to the dipole by the dipole-gradient force, pdg. We set the detuning to the ‘red’(–δ) so that the dipole-gradient force will point in the direction of high light intensity:

pdg =∫ t

0d12

[δ(ω)12�

δ2 + γ 2 + 12�

2

]∂E0(z)∂z

dt (10.151)

Taking into account that � is a function of E0(t), varying from � = 0 before the pulseencounters the dipole to � = �0 at full overlap, and after substituting dt = dz/c, weobtain

pdg = –hδ(ω)2c

ln

[1 +

12�

20

δ2 + γ 2

]� –

12d12E0

c

[δ(ω) 12�0

δ2 + γ 2

]� –

12d12E0

c

[δ(ω) 12�0

δ2 + γ 2 + 12�

20

]

(10.152)

5 The spontaneous emission loss rate of population from the upper state, � = 2γ .6 In Chapter 11 the definition of the on-resonant Rabi frequency is h� = d12E0.


We have added (1/2)�20 to the denominator in the last term on the right so that the

factor in square brackets is the same as in Equations 10.151 and 10.150, and under theassumption that �2

0 << δ2 + γ 2.Now consider F2z, the second term on the right of Equation 10.147. The time inte-

gration can be carried out directly. Using Equation 10.149 and B = (E0/c) cos(ωt – kz),and averaging over an optical cycle, we find the �F2z term contribution to the momentum,

p2 =d12E0

c

[δ(ω) 12�0

δ2 + γ 2 + 12�

20

](10.153)

We see that p2 is twice as great as pdg in absolute value and opposite in sign. As the leadingedge of the pulse passes through the dipole, it receives a positive net kick equivalent to

pnet = pdg + p2 � 12d12E0

c

[δ(ω) 12�0

δ2 + γ 2 + 12�

20

](10.154)

The same result can be obtained from using Equation 10.143. Here the first term, (d ·∇)Ei , does not contribute because there is no z component of the E-field in the incidentpulse. The second term can be regarded as a Lorentz force with d as a bound current.Using Equation 10.149, this Lorentz force can be written as

FL = 2d12 [u cos(ωt – kz) – uω sin(ωt – kz) – v sin(ωt – kz) – vω cos(ωt – kz)]E0

ccos(ωt – kz)

(10.155)where we have set

u =

[δ(ω) 12�

δ2 + γ 2 + 12�

2

]and v =

[γ 1

2�

δ2 + γ 2 + 12�

2

](10.156)

Averaging over an optical cycle and integrating over the leading edge of the pulse7 yields

pL = d12

(uE0

2c– vω

E0

ct)

(10.157)

The first term on the right is the same as Equation 10.153 while the second term expressesthe momentum due to atom recoil absorptive scattering.

Now the energy in the pulse is (1/2)ε0E20V , where V is the pulse spatial volume. The

number of photons in the pulse is (1/2)ε0E20V /hω, and the Lorentz momentum pL per photon

transferred to the dipole is

pLhω

=d12uε0E0V

p0 (10.158)

7 The factor of 1/2 appearing in the first term of Equation 10.157 arises from integrating over thediscontinuity of B at the dipole position.


where p0 = hω/c is the total momentum originating in the light pulse. The momentumremaining in the light pulse after the transfer of pL to the dipole is

pL = p0

(1 –

d12uε0E0V

)� p0

1 + d12uε0E0V

(10.159)

We can identify the susceptibility of the in-phase driving term for a single dipole fromEquation 11.87 in Chapter 11 as

χ ′ =2d12uε0E0V

(10.160)

and therefore,

pL � p01 + 1

2χ′ (10.161)

Finally, we recognise that the real part of the refractive index n =√ε =√1 + χ ′ � 1 + 1/2χ ′.

So that

pL � p0n

(10.162)

which we associate with the Abraham momentum.The field momentum gained by the transfer pdg is

pdg = p0

(1 +

d12uε0E0V

)= p0

(1 +

χ ′

2

)� p0n (10.163)

which we associate with the Minkowski momentum.We now have shown that the Lorentz force of Equation 10.143 results in a transfer of

Abraham momentum to the dipole, and we can identify the dipole gain in momentum aspkin, the kinetic momentum. What about the dipole gradient momentum? With red detun-ing the dipole is pulled by the gradient to a region of higher field intensity, and we haveseen that the momentum transferred to the field is the Minkowski momentum. According toEquation 10.142, the Minkowski momentum gained by the field must be equal to the loss ofcanonical momentum by the dipole. Therefore, we must associate the dipole gradient mo-mentum to the canonical momentum. This association can be rationalised by rememberingthat the canonical momentum operator is given by ih∇, and the dipole-gradient force resultsfrom the field gradient (Equation 10.150).

10.4.15 Conclusions

The Einstein thought experiment establishes the necessity of momentum in the electro-magnetic field, and there can be no doubt that the Abraham momentum and Minkowskimomentum are valid expressions. There is almost universal agreement that the Poynting vec-tor, S = E×H, describes the energy flux of the electromagnetic field both in free space and inmaterials. The momentum in free space is therefore given byG = (1/c2)S. The question then

Summary 297

is: what is the momentum of the electromagnetic field propagating within ponderable media?The Balazs thought experiment says it must be the Abraham momentum, although anotherthought experiment due to Padgett [35] concludes equally convincingly that in the case of dif-fraction it should be the Minkowski momentum. Appeal to experiment is not as conclusive asone might think because close analysis of the submerged-mirror project reveals that the forcemeasured on the mirror is not just the change of the electromagnetic momentum upon reflec-tion in the medium, but contains a force term due to the material itself acting on the mirror; ormay depend on the phase of the reflection. At this writing (early 2016), experiments requiredto decide this issue have not yet been carried out. The suggestion that both forms of the elec-tromagnetic momentum may be valid under different circumstances is appealing, but rigor-ous criteria specifying the use of the Abraham or Minkowski form have yet to be worked out.Furthermore, specifying the form of the canonical momentum in the Coulomb gauge meansthat the Minkowski-canonical association is not gauge invariant. The Abraham form seems tobe definitely associated with Newtonian kinetic momentum, but the conditions appropriate toassociating the Minkowski form with the canonical momentum are more difficult to pin down(interference, diffraction, intensity gradients?). The BEC experiment uses a standing-wavetwo-field Ramsey interferometer with the individual atoms scattering off the gradients of theperiodic light potential. According to theMinkowski-canonical association it is not surprising,therefore, that the Minkowski momentum appears. At this time, theory appears to favour theAbraham form of electromagnetic momentum in ponderable media when light forces due tofield gradients are not present, and the Minkowski form when they are. A way forward in thesubmerged-mirror experiments is to repeat them with a series of low-loss reflectors of varyingphase.

10.5 Summary

This chapter treats the question of how momentum is transported from a light source toand through material objects by propagating light fields. The idea of what form momen-tum might take in fields is first established by the Einstein thought experiment. The Balazsthought experiment then concludes that momentum must take the Abraham form as lightpropagates through dielectrics with refractive index greater than vacuum. A second thoughtexperiment based on reflection, however, seems to indicate that the momentum, the rate ofchange of which gives rise to the radiative force on a mirror, must take the Minkowski form.The question of radiative force and force laws is then opened and examined. Maxwell’s equa-tions can be rearranged to fit various physical interpretations. The Lorentz model expressesMaxwell’s equations entirely in terms of the fields E and B, and suggests that the sourceof material magnetisation consists of a density of Amperian current loops. The magneticdipole model expresses the same Maxwell equations in terms of E and H, but the interpret-ation of bound magnetic charge and bound magnetic current invites a picture of magneticdipole density analogous to the electric dipole density of the polarisation field. These twomodels of the constitution of magnetic matter give rise to two different expressions for theenergy flux density. The two are linked by Equation 10.54. In contrast to energy conser-vation, momentum conservation must be expressed in terms of the divergence of a tensor,not a vector field. For readers not yet initiated into the mysteries of second-rank tensors,we allow a digression into the exploration of their properties before returning to Lorentzand Einstein–Laub force laws. The next topic is a somewhat cursory discussion of hidden


energy and hidden momentum, the motivation being the clarification of issues without be-ing drawn into a debilitating controversy. After the discussion of hidden energy and hiddenmomentum, the focus passes to key experiments that have attempted to answer the ques-tion of the correct form of momentum in light fields within ponderable media. Light forcesapplied to mirrors submerged in a series of liquids of refractive index above unity indicatesthat the momentum varies as the refractive index of the material. The experimental find-ing supports the Minkowski form, but according to Mansuripur and Kemp, the issue is farfrom settled because either a phase difference at the reflection plane or an extra force term atthe dielectric-mirror interface itself may contribute to the apparent light force on the mirror.A second experiment involving a Bose–Einstein condensate (BEC) and a two-field Ramseyinterferometer setup is not really any more successful at finding a definitive, once-and-for-allanswer. Although again, the measurement supports the Minkowski form, this result may bea consequence of the close-to-unity refractive index of the BEC. The chapter ends with theproposition of Barnett that the ‘correct’ form for the momentum depends on the specificcircumstances of the measurement, and relies on drawing a fundamental distinction betweenkinetic and canonical forms of the momentum. An example of resonant light interacting witha two-level atom coupled by a dipole interaction illustrates how the kinetic momentum can beassociated with the Abraham form and the canonical momentum can be associated with theMinkowski form.


1. D. J. Griffiths, Introduction to Electrodynamics, 3rd edition, Chapters 4–9, Pearson-Addison-Wesley (1981).

2. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media,Chapter IX, 2nd edition, Elsevier (1982).

3. J. A. Stratton, Electromagnetic Theory, Chapters I and II, McGraw-Hill (1941).

4. J. D. Jackson, Classical Electrodynamics, Chapter 6, John Wiley & Sons (1962).

5. M. Mansuripur, Field, Force, Energy, and Momentum in Classical Electrodynamics, Chap-ters 2 and 10, Bentham Books (2011).

6. I. S. Sokolnikoff and R. M. Redheffer,Mathematics of Physics and Modern Engineering, g4, McGraw-Hill (1958).

7. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Processus d’interaction entrephotons et atomes, Chapter 5, Editions du CNRS (1988).

10.7 Bibliography

[1] J. P. Gordon, Radiation Forces and Momenta in Dielectric Media. Phys Rev A: At MolOpt Phys vol 8, pp. 14–21 (1973).

[2] N. Balazs, The Energy-Momentum Tensor of the Electromagnetic Field inside Matter.Phys Rev vol 91, pp. 408–411 (1953).

[3] M. Mansuripur, Momentum exchange effect, Nat Photonics vol 7, pp. 765–766(2013).

[4] A. Ashkin and J. M. Dziedzic, Radiation Pressure on a Free Liquid Surface. Phys RevLett vol 30, pp. 139–142 (1973).

Bibliography 299

[5] R. V. Jones and B. Leslie, The measurement of optical radiation pressure in dispersivemedia. Proc R Soc London Ser A vol 360, pp. 347–363 (1978).

[6] G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle,and D. E. Pritchard, Photon Recoil Momentum in Dispersive Media. Phys Rev Lettvol 94, pp. 170,403 (2005).

[7] S. M. Barnett, Resolution of the Abraham-Minkowski Dilemma. Phys Rev Lettvol 104, pp. 070,401 (2010).

[8] M. Mansuripur, Resolution of the Abraham-Minkowski controversy. Opt Communvol 283, pp. 1997–2005 (2010).

[9] B. A. Kemp, Resolution of the Abraham-Minkowski debate: Implications for theelectromagnetic wave theory of light in matter. J Appl Phys vol 109, pp. 111,101(2011).

[10] M. Mansuripur, On the Foundational Equations of the Classical Theory of Electro-dynamics. Resonance vol 18, pp. 130–155 (2013).

[11] M. Mansuripur, The Force Law of Classical Electrodynamics: Lorentz versus Einsteinand Laub. Proc SPIE vol 8810, pp. 88,100K (2013).

[12] M. Mansuripur, Nature of electric and magnetic dipoles gleaned from the Poyntingtheorem and the Lorentz force law of classical electrodynamics. Opt Commun vol 284,pp. 594–602 (2011).

[13] B. A. Kemp,Macroscopic Theory of Optical Momentum. Prog Optics vol 60, pp. 437–488 (2015).

[14] A. Einstein and J. Laub, Über die im elektromagnetischen Felde auf ruhende Körperaus geübten ponderomotorischen Kräte. Ann Phys vol 331, pp. 541–550 (1908).

[15] D. F. Nelson, Momentum, pseudomomentum, and wave momentum: To-ward resolving the Minkowski-Abraham controversy. Phys Rev vol 44,pp. 3985–3996 (1991).

[16] P. W. Milonni and R. W. Boyd,Momentum of Light in a Dielectric Medium. Adv OptPhotonics vol 2, pp. 519–553 (2010).

[17] D. J. Griffiths, Resource Letter EM-1: Electromagnetic Momentum. Am J Phys vol 80,p. 7 (2012).

[18] C. Baxter and R. Loudon, Radiation pressure and the photon momentum in dielectrics.JMod Opt vol 57, pp. 830–842 (2010).

[19] S. M. Barnett and R. Loudon, The enigma of optical momentum in a medium. PhilosTrans R Soc London Ser A vol 368, pp. 927–939 (2010).

[20] P. L. Saldanha, Division of the energy and of the momentum of electromagnetic wave inlinear media into electromagnetic and material parts. Opt Commun vol 284, pp. 2653–2657 (2011).

[21] M. Mansuripur, A. R. Zakharian, and E. M. Wright, Electromagnetic-force distribu-tion inside matter. Phys Rev A: At Mol Opt Phys vol 88, pp. 023,826 (2013).

[22] W. Shockley and R. P. James, ‘Try Simplest Cases’ Discovery of ‘Hidden Momentum’Forces on ‘Magnetic Currents’. Phys Rev Lett vol 18, pp. 876–879 (1967).

[23] R. V. Jones and J. C. S. Richards, The pressure of radiation in a refracting medium.Proc R Soc London. Ser A vol 221, pp. 480 (1954).


[24] M. Mansuripur, Radiation Pressure on Submerged Mirrors: Implicationsfor the Momentum of Light in Dielectric Media. Opt Express vol 15,pp. 2677–2682 (2007).

[25] B. A. Kemp and T. M. Grzegorczyk, The observable pressure of light in dielectricfluids. Opt Lett vol 36, pp. 493–495 (2011).

[26] M. Mansuripur. Personal communication (2015).[27] M. Mansuripur and A. R. Zakharian, Whence the Minkowski momentum. Opt

Commun vol 283, pp. 3557–3563 (2010).[28] R. Loudon, Theory of the radiation pressure on dielectric surfaces. JMod Opt vol 49,

pp. 821 (2002).[29] M. Mansuripur, Radiation pressure and the linear momentum of the electromagnetic

field. Opt Express vol 12, pp. 5375–5401 (2004).[30] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Massachusetts

(1950).[31] S. M. Barnett and R. Loudon, On the electromagnetic force on a dielectric medium.

J Phys B: At Mol Opt Phys vol 39, pp. S671–S684 (2006).[32] A. R. Zakharian, P. Polynkin, M. Mansuripur, and J. V. Moloney, Single-beam

trapping of micro-beads in polaized light: Numerical simulations. Opt Express vol 14,pp. 3660–3676 (2006).

[33] E. A. Hinds and S. M. Barnett, Momentum Exchange between Light and a SingleAtom: Abraham or Minkowski? Phys Rev Lett vol 102, pp. 050,403 (2009).

[34] S. M. Barnett. Personal communication (2015).[35] M. J. Padgett, On diffraction within a dielectric medium as a example of the Minkowski

formulation of optical momentum. Opt Express vol 16, pp. 20,864–20,868 (2008).

11

Atom-Light Forces

11.1 Introduction

As we have seen in Chapter 10, a light beam carries momentum, and light scatteringtransfers some or all of that momentum to a ponderable object. The time rate of changeof momentum produces a force on the object. This property of light was first dem-onstrated through the observation of a very small transverse deflection (3 × 10–5 rad)in a beam of sodium atoms exposed to light from a resonance lamp. A single-modelaser source tuned to the atomic resonance transitions greatly facilitates the deflectionmeasurement due to spectral purity, intensity, and directionality of the laser beam. Al-though these results kindled interest in using light forces to control the motion of neutralatoms, the basic groundwork for the understanding of light forces acting on atoms wasnot laid out before the end of the 1970s. Application of light forces to atom coolingand trapping was not accomplished before the mid-1980s. In this chapter we discusssome fundamental aspects of light forces and schemes employed to cool and trap neutralatoms.

The light force exerted on an atom can be of two types: a dissipative, spontaneous forceand conservative, dipole force1. The spontaneous force arises from the impulse experi-enced by an atom when it absorbs or emits a quantum of photon momentum. When anatom scatters light, the resonant scattering cross section can be written as

σ0a =g1g2

πλ20

2(11.1)

where λ0 is the resonant wavelength. In the optical region of the spectrum wavelengthsare on the order of a few hundred nanometres, so resonant scattering cross sectionsbecome relatively large, ∼10–9 cm2. Each photon absorbed transfers a quantum ofmomentum hk to the atom in the direction of the light beam propagation. Spontan-eous emission following the absorption occurs in random directions, and over manyabsorption-emission cycles, it averages to zero. As a result, the net spontaneous force

1 In this chapter we drop the second term in Equation 10.146 of Section 10.4.13 since in the usualrealisations of light forces on atoms this term is negligible compared to the field-gradient term.


302 Atom-Light Forces

hk

pa = mva

pa = hk + mva pa = hk + mva

Figure 11.1 Left: atom moves to the right with mass m, velocity va, andabsorbs a photon propagating to the left with momentum hk. Centre: excitedatom experiences a change in momentum pa = mva + hk. Right: photonisotropic re-emission results in an average momentum change for the atom, aftermultiple absorptions and emissions, of 〈pa〉 = mva + hk.

acts on the atom in the direction of the light propagation, as shown schematically inFigure 11.1. The saturated rate of photon scattering by spontaneous emission (the re-ciprocal of the excited-state lifetime) fixes the upper limit to the force magnitude. Theforce is often called the radiation pressure force.

The dipole-gradient force can be readily understood by considering the light as aclassical wave. It is simply the time-averaged force arising from the interaction of thetransition dipole, induced by the oscillating electric field of the light, with the spatialgradient of the electric field amplitude (see, however, Section 10.4.13 of Chapter 10).Focusing the light beam controls the magnitude of this gradient, and detuning the opticalfrequency below or above the atomic transition controls the sign of the force acting onthe atom. Tuning the light below resonance attracts the atom to the region of maximumintensity (usually along the light beam longitudinal axis), while tuning above resonancerepels the atom from this region. The dipole force is a stimulated process in which nonet exchange of energy between the field and the atom takes place. Photons are absorbedfrom one mode and reappear by stimulated emission in another. Momentum conserva-tion requires that the change of photon propagation direction from initial to final modeimparts a net recoil to the atom. Unlike the spontaneous force, there is, in principle, noupper limit to the magnitude of the dipole force since it is a function only of the fieldgradient and detuning. This statement is strictly true only for two-level atoms. In realatoms or molecules, excited upper states or the ionisation or dissociation continuum willeventually couple into the optical interaction.

11.2 The atom as a damped harmonic oscillator

In order to bring these introductory comments into focus, we first discuss atomic di-pole emission as a classical damped harmonic oscillator. In Section 2.8 we considered aclassical electric dipole aligned along the z-axis and wrote the dipole as

The atom as a damped harmonic oscillator 303

p(t) = q0a cosωtz

If we consider an atom as an electron bound to a massive proton, subject to a radialrestoring force centred at the proton position, then we can write down a similar dipoleexpression,

p(t) = –er(t) p0 = –er0 (11.2)

where –e is the electron charge, r is the radial distance from the atom centre, and r0 isthe maximum radial distance. The electron is pictured as vibrating radially through theproton position and in this simple model exhibits no angular momentum from orbitalmotion about the charge centre. Using the result for power emitted by a classical dipole,Equation 2.157, we can write the optical-cycle-averaged power emitted by the atom as

〈W 〉 = 14πε0

ω40p

20

3c3=

e2

4πε0

ω40r

20

3c3(11.3)

For harmonic oscillation the maximum acceleration is a0 = –ω20r0 and

〈W 〉 = e2

4πε0

a203c3

(11.4)

Writing the cycle-averaged acceleration as 〈a · a〉 = ⟨a2⟩ = (1/2)a20, we have

〈W 〉 = e2

4πε0

2⟨a2⟩

3c3(11.5)

and we see that the average power emitted by the oscillating electron is proportional tothe average square of the electron acceleration. A charged particle must be acceleratingto emit radiation; an electron moving at constant velocity, not subject to external forces,does not emit.

The energy (cycle-averaged) emitted by the oscillator must come from mechanical(kinetic plus potential) energy, Emech:

Emech =12meω

20r

20 (11.6)

where me is the electron mass. The average power emitted by the oscillator can beequated to the diminution of mechanical energy with time:

〈W 〉 = –Emech

τ(11.7)


From Equations 11.3, 11.6, and 11.7 we see that

τ =(

e2

4πε0

)–1 3mec3

2ω20

(11.8)

The ‘lifetime’ of the oscillator τ can be written a little more simply by introducing theclassical radius of the electron,

rc =e2

4πε0

1mec2

� 2.8× 10–15 m (11.9)

and

τ =3c

2rcω20

(11.10)

In differential form, Equation 11.7 is written as

〈W 〉 = –dEmech

dt

so

dEmech

Emech= –

dtτ

(11.11)

and

Emech(t) = Emech(0)e–t/τ (11.12)

We see that the internal energy of the atom, modelled as a classical electron oscillat-ing around a positive charge centre, decreases exponentially with time as the energy isradiated away.

A specific example provides some feeling for the relevant time and energy scales.Consider a sodium atom, structured with 10 electrons in a ‘closed shell’ and one elec-tron outside. This exposed or ‘active’ electron is subject to an effective positive chargeat the nucleus roughly equal to that of one proton. The potential and electric field ex-perienced by the active electron is radial and the simple dipole oscillator model mighttherefore be appropriate. Emission from the first excited state to the ground state is con-centrated in a narrow band of wavelengths centred at 590 nm, the well-known ‘D line’of atomic spectroscopy. Considering this emission as emanating from a dipole oscillatorwith resonant frequency ω0,

ω0 =2πcλ

= 3.2× 1015 s–1

Radiative damping and electron scattering 305

and

τ =3c

2rcω20

� 16× 10–9 s

This result is actually quite close to the measured lifetime of the first excited state ofthe sodium atom and lends credence to the harmonic oscillator model. In reality, how-ever, the atom must be treated quantum mechanically because there is no lower limitto the classical radiative emission, and the classical electron would fall into the nucleusas its mechanical energy diminishes to zero. Furthermore, the first excited state witha quantum-mechanically ‘allowed’ transition possesses one quantum of orbital angularmomentum, and this angular momentum does not appear in the simple oscillator model.

11.3 Radiative damping and electron scattering

Nevertheless, the damped harmonic oscillator model provides some insight into thephysics of atomic emission, absorption of radiation, and atomic scattering. We canwrite down the equation of motion of the classical, radiatively damped, externally drivenharmonic oscillator, moving along the z-axis, as

d2zdt2

+1τ

dzdt

+ ω20z = –

eEzme

e–iωt (11.13)

where as before ω0 is the characteristic frequency of the oscillator, e and me are theelectron charge and mass, respectively, τ the classical oscillator lifetime, and Ez is theamplitude of the external driving field oscillating at frequency ω. The solution to thisequation of motion is

z =–eEze–iωt

me[(ω20 – ω

2)– iω/τ

] (11.14)

From this result we can write down the frequency-dependent dipole moment,

p = –ezz =–eEze–iωt

me[(ω20 – ω

2)– iω/τ

] z (11.15)

From Equation 11.3 we have, for the average dissipated power,

〈W 〉 = e4E2zω

4

12πε0m2e

[(ω20 – ω

2)2 + (ω/τ)2

] (11.16)


Using the classical electron radius, Equation 11.9, we can write this expression for thedissipated power as the product of an energy flux and a scattering cross section,

〈W 〉 =(ε0E2

z

2

)c︸︷︷︸

energy flux

·{[

8πr2c3

][ω4(

ω20 – ω

2)2

+ (ω/τ)2

]}︸︷︷︸

cross section

(11.17)

where the scattering cross section is defined as

σ (ω) =[8πr2c3

][ω4(

ω20 – ω

2)2

+ (ω/τ)2

](11.18)

11.4 The semiclassical two-level atom

We have seen in Chapter 6 that we can infer a purely phenomenological relation be-tween stimulated and spontaneous emission by invoking the Einstein rate expression,Equation 6.9. We found that, assuming thermal equilibrium between states 1, 2, therate constants of stimulated absorption and emission, B12,B21 are equal, and the rateof spontaneous emission is related to B21 by Equation 6.15:

A21

B21=hω3

0

π2c3

We will now develop a quantum semiclassical expression for the stimulated rate of ab-sorption and relate it back to B21 and A21. Then we will interpret this semiclassical resultin terms of a classical radiating dipole.

11.4.1 Coupled equations of a two-state system

We start with the time-dependent Schrödinger equation,

H�(r, t) = ih�

dt(11.19)

and write the stationary-state solution of level n as

�n(r, t) = ψn(r)e–iEnt/h = ψn(r)e–iωnt (11.20)

The time-independent Schrödinger equation then becomes

HAψn(r) = Enψn(r) (11.21)

The semiclassical two-level atom 307

where the subscript A indicates ‘atom’. Then for the two-level system we have

HAψ1 = E1ψ1 = hω1ψ1

HAψ2 = E2ψ2 = hω2ψ2

and write

hω0 ≡ h (ω2 – ω1) = E2 – E1 (11.22)

Now we add a time-dependent term to the Hamiltonian that will turn out to beproportional to the oscillating classical field with frequency not far from ω0:

H = HA + V (t) (11.23)

With the field turned on, the state of the system becomes a time-dependent linearcombination of the two stationary states

�(r, t) = C1ψ1e–iω1t +C2(t)ψ2e–iω2t (11.24)

which we require to be normalised,∫|�(r, t|2 dτ = |C1(t)|2 + |C2(t)|2 = 1 (11.25)

Now, if we substitute the time-dependent wave function (Equation 11.24) back intothe time-dependent Schrödinger equation (Equation 11.19), multiply on the left withψ∗1 e

iω1t, and integrate over all space, we get

C1

∫ψ∗1 Vψ1 dr +C2e–iω0t

∫ψ∗1 Vψ2 dr = ih

dC1

dt(11.26)

From now on we will denote ‘matrix elements’∫ψ∗1 Vψ1 dr and

∫ψ∗1 Vψ2 dr as V11 and

V12, so we have

C1V11 +C2e–iω0tV12 = ihdC1

dt(11.27)

and similarly for C2, we obtain

C1eiω0tV21 +C2V22 = ihdC2

dt(11.28)

These two coupled equations define the quantum-mechanical expression, and their so-lutions, C1 and C2, define the time evolution of the state wave function, Equation 11.24.Of course, any measurable quantity is related to |�(r, t)|2; consequently we are reallymore interested in |C1|2 and |C2|2 than the coefficients themselves.


11.4.2 Field coupling operator

A single-mode radiation source, such as a laser, aligned along the z-axis, will produce anelectromagnetic wave with amplitude E0, polarisation e, and frequency ω:

E = eE0 cos(ωt – kz) (11.29)

with the magnitude of the wave vector

k =2πλ

and ω = 2πν = 2πcλ= kc (11.30)

An optical wavelength in the visible region of the spectrum, say, λ = 600nm� 1.1 ×104 a0, is evidently several orders of magnitude greater than the characteristic atomicscale (� a0). Therefore, over the spatial extent of the atom-field interaction, the kz termin Equation 11.29 is negligible, and we can consider the field amplitude to be constantover the scale length of the atom. We can make, therefore, the dipole approximation inwhich the leading interaction term between the atom and the optical field is the scalarproduct of the atom dipole d, defined as

d = –er = –e∑j

rj (11.31)

and the field coupling operator is

V = –d · E (11.32)

The operator V has odd parity with respect to the electron coordinate r so that the matrixelements V11 and V22 vanish, and only atomic states of opposite parity can be coupledby the dipole interaction. The explicit expression for V12 is

V12 = eE0r12 cosωt (11.33)

with

r12 =∫ψ∗1

⎛⎝∑

j

rj · e⎞⎠ψ2 dτ (11.34)

Equation 11.34 describes the matrix element of the electronic coordinate vector operatorsummed over all electrons (assuming a two-state, multi-electron atom) and projectedonto the E-field direction of the optical wave. The transition dipole moment matrixelement is defined as

p12 ≡ er12 (11.35)


It is convenient to collect all these scalar quantities into one term �0 with units offrequency, often called the ‘on-resonance Rabi frequency’:

�0 ≡ p12E0

h=eE0r12h

(11.36)

So finally we have

V12 = h�0 cosωt (11.37)

11.4.3 Calculation of B12

Now we rewrite Equations 11.27 and 11.28 in terms of the Rabi frequency:

�0 cosωte–iω0tC2 = idC1

dt(11.38)

�∗0 cosωteiω0tC1 = i

dC2

dt(11.39)

We take the initial conditions to be C1(t = 0) = 1 and C2(t = 0) = 0 and recall that|C2(t)|2 expresses the probability of finding the population in the excited state at time t.The time rate of probability increase, of finding the system in state 2, is given by

|C2(t)|2t

(11.40)

This expression can be set equal to the stimulated absorption term in the Einsteinrelation Equation 6.9:

|C2(t)|2t

= B12ρ(ω)dω (11.41)

Now we seek C2(t) from Equation 11.39 and the initial conditions. In the weak-fieldregime where only terms linear in �0 are important, we have

C2(t) =�∗02

[1 – ei(ω0+ω)t

ω0 + ω+

1 – ei(ω0–ω)t

ω0 – ω

](11.42)

If the frequency ω of the driving wave approaches the transition resonant frequencyω0, the exponential in the first term in brackets will oscillate at about twice the atomicresonant frequency ω0(∼1015 s–1), very fast compared to the characteristic rate of weak-field optical coupling (∼108 s–1). Therefore, over the time of the transition, the firstterm in Equation 11.42 will be negligible compared to the second. To a quite goodapproximation we can write,


C2(t) � �∗02

[1 – ei(ω0–ω)t

ω0 – ω

](11.43)

This expression is sometimes called the rotating wave approximation (RWA) because itcorresponds to the solution for C2 in a coordinate frame rotating at frequency ω0. FromEquation 11.43 we have

|C2(t)|2 = |�0|2 sin2 [(ω0 – ω) t2

](ω0 – ω)2

(11.44)

In order to arrive at a useful expression relating |C2(t)|2 to the Einstein B coefficients,we have to take into account the fact that there is always a finite width in the spectraldistribution of the excitation source. The source might be, for example, an incoherentbroadband arc lamp, the output from a monochromator coupled to a synchrotron, or anarrowband mono-mode laser whose spectral width would probably be narrower thanthe natural width of the atomic transition. So if we write the field energy as an integralover the spectral energy density of the excitation source in the neighbourhood of thetransition frequency,

12ε0E2

0 =∫ ω0+

12�ω

ω0–12�ω

ρω dω (11.45)

where the limits of integration, ω0 ± 12�ω, refer to the spectral width of the excitation

source, and recognise from Equation 11.43 that

|C2(t)|2 =(eE0r212εoh

)2 sin2[(ω0 – ω) t2

](ω0 – ω)2

(11.46)

we can then substitute Equation 11.45 into Equation 11.46 to find that

|C2(t)|2 = e22r212ε0h2

∫ ω0+12�ω

ω0–12�ω

ρωsin2

[(ω0 – ω) t2

](ω0 – ω)2

dω (11.47)

For conventional broadband excitation sources we can safely assume that the spectraldensity is constant over the line width of the atomic transition and take ρω outside theintegral operation and set it equal to ρ(ω0). Note that this approximation is not valid fornarrow band mono-mode lasers. Let us assume a fairly broadband continuous excitationso that t(ω0 – ω)� 1. In this case

∫ ω0+12�ω

ω0–12�ω

ρωsin2

[(ω0 – ω) t2

](ω0 – ω)2

dω =π t2

(11.48)


and the expression for probability of finding the atom in the excited state becomes

|C2(t)|2 =e2πr212ε0h2

ρ(ω0)t (11.49)

Remembering that Equation 11.41 provides the bridge between the quantum and clas-sical expressions for the excitation rate, we can now write the Einstein B coefficient interms of the transition moment as

B12ρ(ω0) =|C2(t)|2

t=e2πr212ε0h2

ρ(ω0) (11.50)

or

B12 =e2πr212ε0h2

(11.51)

The square of the transition moment matrix element r212 averaged over all spatialorientations is

⟨|r12|2⟩ = r212⟨cos2 θ

⟩=

13r212 (11.52)

where θ is the angle between the transition moment and the E-field of the plane waveexcitation. We have, finally,

B12 =e2πr2123ε0h2

(11.53)

and from the definition of the dipole moment p12, Equation 11.35:

B12 =πp2123ε0h2

(11.54)

Assuming that states 1 and 2 are not degenerate, B12 = B21, and we have from Equa-tion 6.15 an expression for the spontaneous emission rate in terms of the dipole transitionmoment calculated from our two-level model:

A21 =ω30p

212

3πε0hc3(11.55)

11.4.4 Spontaneous emission as loss

Everything we have developed in Section 11.4 up to this point involves the coupling ofone optical field mode to a two-level atom. The Schrödinger equation is adequate to


describe the time evolution of this system because it can always be expressed by a pure-state wave function. Spontaneous emission, however, cannot be characterised by the timeevolution to a final-state wave function, but only as a probability distribution of final-statewave functions. The time evolution of a probability distribution requires a density matrixdescription, and the governing equation of motion is the Liouville equation. Nevertheless,within the two-level picture we can treat spontaneous emission as a phenomenological,dissipative loss term, and thereby avoid a density matrix treatment that would carry usbeyond the scope of this section.

We take account of this dissipative loss by modifying Equation 11.39 to include aradiative loss rate constant γ :

�∗0 cosωteiω0tC1 – iγC2 = i

dC2

dt(11.56)

If the driving field (�∗ = 0) is turned off at t = t0,

– iγC2 = idC2

dt(11.57)

and

C2(t) = C2(t0)e–γ (t–t0) (11.58)

So the probability of finding the atom in the excited state is

|C2(t)|2 = |C2(t0)|2 e–2γ t (11.59)

In an ensemble of N atoms in a volume, the number N2 in the excited state 2 is

N2 = N |C2(t)|2 = N2(t0)e–2γ t (11.60)

If we compare this behaviour to the result obtained from the Einstein rate equation, wesee that

A21 = 2γ ≡ � (11.61)

Now the steady-state solution for our new, improved C2(t) coefficient (Equa-tion 11.42) is

C2(t) = –12�∗0

[ei(ω0+ω)t

ω0 + ω – iγ+

ei(ω0–ω)t

ω0 – ω – iγ

](11.62)


Using Equations 11.24 and 11.34 we can write the net transition dipole 〈p12〉 bysumming over the electronic coordinate vectors of our multi-electron, two-level atom:

〈p12〉 = –e∫�∗

∑j

rj�dτ = –e 〈r12〉 (11.63)

and the time dependence of the atomic transition dipole moment in terms of the timedependence of C2(t):

〈p12〉 = –e[C∗1C2(t) 〈r12〉 e–iω0t +C2(t)∗C1 〈r21〉 eiω0t

](11.64)

Substituting C2(t) from Equation 11.62 and assuming a weak-field excitation (C1 � 1)we can then write

〈p12(t)〉 = e2 |r12|2 E0

2h

[eiωt

ω0 + ω – iγ+

e–iωt

ω0 – ω – iγ+

e–iωt

ω0 + ω + iγ+

eiωt

ω0 – ω + iγ

](11.65)

11.4.5 Polarisation and transition dipole moment

The driving field E(t) can also be expressed in terms of positive and negative frequencyterms:

E(t) = E0 cosωt =12E0[eiωt + e–iωt

](11.66)

and from Equation 2.17 the polarisation field in extended matter can be written as

P(t) =12ε0E0

[χ(ω)e–iωt + χ(–ω)eiωt

](11.67)

with χ(±ω), the susceptibility. Now, as we have seen in Chapter 4, Sections 4.2 and 4.3,the polarisation vector field can be considered a dipole density:

P(t) =NV〈p12(t)〉 = n 〈p12(t)〉 (11.68)

where N are the number of dipoles in volume V and n is the number density. Afterreplacing |r12|2 with its orientation averaged value, 1

3 |r12|2, we have for the polarisationvector,

P(t) =n |〈p12〉|2

6hE0

[(1

ω0 – ω – iγ+

1ω0 + ω + iγ

)e–iωt+

(1

ω0 + ω – iγ+

1ω0 – ω + iγ

)eiωt]

(11.69)


Comparison of Equations 11.67 and 11.69 allows us to write the susceptibility in termsof the transition dipoles:

χ(ω) =n |〈p12〉|23ε0h

(1

ω0 – ω – iγ+

1ω0 + ω + iγ

)(11.70)

Identify the real χ ′ and imaginary parts χ ′′ of the susceptibility, χ = χ ′(ω) + iχ ′′(ω):

χ(ω) =n |〈p12〉|23ε0h

[(ω0 – ω

(ω0 – ω)2 + γ 2+

ω0 + ω(ω0 + ω)2 + γ 2

)

+ iγ(

1(ω0 – ω)2 + γ 2

–1

(ω0 – ω)2 + γ 2

)](11.71)

and apply the RWA:

χ ′(ω) =n |〈p12〉|23ε0h

(ω0 – ω

(ω0 – ω)2 + γ 2

)(11.72)

χ ′′ =n |〈p12〉|23ε0h

(γ

(ω0 – ω)2 + γ 2

)(11.73)

We are usually interested in frequencies ω tuned not too far from ω0, and the term‘detuning’ is often used to denote the frequency difference,

�ω = ω – ω0 (11.74)

This choice for �ω means that ‘blue detuning’ (ω > ω0) is positive and ‘red detuning’(ω < ω0) is negative. With this convention in mind, and remembering that � = 2γ(Equation 11.61), we write the real and imaginary parts of the susceptibility as

χ ′(ω) = –n |〈p12〉|23ε0h

(�ω

�ω2 +(�2

)2)

(11.75)

χ ′′(ω) =n |〈p12〉|23ε0h

(�/2

�ω2 +(�2

)2)

(11.76)

Around the resonance frequency ω0, the real part of the susceptibility χ ′ exhibits a ‘dis-persive’ profile. Right on resonance χ ′ is zero. It is positive to the red of the resonanceand negative to the blue, falling off with the square of the detuning. In contrast, χ ′′ ex-hibits an ‘absorptive’ profile. It is positive and peaks at resonance, falling off with thesquare of the detuning on each side. Figures 11.2 and 11.3 trace the two susceptibilityterms around the detuning resonance.


−50 −40 −30 −20 −10 0 10 20 30 40 50−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

detuning (Γ/2 units)

χ′ (

arbi

trar

y un

its)

Figure 11.2 Dispersive profile of the real part of the susceptibility χ ′ plotted against detuning �ω.

From Equation 11.71 we see that χ ′ is symmetric with respect to the interchange ofω and –ω while χ ′′ is antisymmetric:

χ ′(ω) = χ ′(–ω) (11.77)

χ ′′(ω) = –χ ′′(–ω) (11.78)

Substituting these symmetry relations into Equation 11.67 allows us to separate the timedependence of real and imaginary parts of the susceptibility:

P(t) = ε0E0(χ ′ cosωt + χ ′′ sinωt

)(11.79)

The time dependence of the polarisation field shows that the dispersive term is drivenby the ‘in-phase’ real part of the E-field cosωt while the absorptive term is driven by the‘in-quadrature’ sinωt.


−50 −40 −30 −20 −10 0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

χ′′(a

rbitr

ary

units

)

Detuning (Γ/2 units)

Figure 11.3 Absorptive profile of the imaginary part of the susceptibility χ ′′ plotted againstdetuning �ω.

Using the real form of the driving field E = E0 cosωt, the interaction energy betweenthe driving E-field and the polarisation field is

E = –P(t) · E(t) = –ε0E0[χ ′ cosωt + χ ′′ sinωt

] · E0 [cosωt] (11.80)

and averaging over a optical cycle we have

〈E 〉 = –12ε0E2

0χ′ (11.81)

The dipole-field interaction energy separates into a real, dispersive part associated withχ ′ and an imaginary, absorptive part associated with χ ′′.

The dipole-gradient and radiation pressure forces 317

11.5 The dipole-gradient and radiation pressure forces

Light interacting with the two-level atom also gives rise to forces. A light beam carriesmomentum, and scattering from an atom produces a transfer of momentum resulting ina force on the atom, F = dp/dt. The light force exerted on an atom can be of two types:a conservative dipole-gradient force and a dissipative spontaneous force.

11.5.1 Dipole-gradient force

The dipole-gradient force can be readily understood by considering the light as a clas-sical wave. It is simply the optical-cycle-averaged force arising from the interaction ofthe transition dipole, induced by the oscillating electro-magnetic field, with the spatialgradient of the electric field amplitude. The dipole-gradient force does not apply topropagating plane waves because the amplitudes are constant. The light must be focusedor spatially modulated in some way. Focusing the light beam controls the gradient of theelectric field amplitude; detuning the frequency below or above the atomic transitioncontrols the sign of the force. Tuning the light to the red of the resonance frequency re-sults in a net attractive force on the atom towards the intensity maximum. Blue-detunedlight pushes the atom to the intensity minimum. The dipole-gradient force is a stimulatedprocess in which no net exchange of energy between the field and the atom takes place.There is, in principle, no upper limit to the magnitude of the dipole-gradient force sinceit is a function only of the field gradient and detuning. The general expression for lightforces is given by the scalar product of the polarisation field and the gradient operatingon the applied E-field:

F = (P ·∇)E (11.82)

F = P ·∇ [E0(r) cos(k · r – ωt)] (11.83)

F = ε0χE ·∇ [E0(r) cos(k · r – ωt)] (11.84)

F = ε0E0[χ ′(ω) cosωt + χ ′′(ω) sinωt

] ·∇ [E0(r) cos(k · r – ωt)] (11.85)

The gradient operator acts on the product of the field amplitude and phase, resulting intwo terms:

∇ [E0(r) cos(k · r – ωt)] = ∇E0(r) cos(k · r – ωt) + E0(r)∇ cos(k · r – ωt) (11.86)

The first of these terms gives rise to the dipole-gradient force and the second to theradiation pressure force. Substituting the first term on the right of Equation 11.86 intoEquation 11.85, and taking the optical-cycle average, results in

Fdg =12ε0E0∇E0(r)χ ′ (11.87)

Fdg =14ε0∇E0

2(r)χ ′ (11.88)


In writing Equations 11.87 and 11.88 we have made use of the fact that k · r is negligibleover the spatial extent of the atom (r ∼ 2a0), where a0 is the Bohr radius, the atomicunit of length. Now substituting Equation 11.75 for χ ′ into 11.88:

Fdg =14ε0∇E0

2(r)

[–|〈p12〉|23ε0h

(�ω

�ω2 +(�2

)2)]

(11.89)

We see that in the expression for Fdg the χ ′ factor includes a negative sign, but due to ourconvention for �ω, Equation 11.74, the detuning to the red is also negative. The dipolegradient force with red detuning points therefore to a higher intensity gradient, whileblue detuning results in a ‘repulsive’ force, away from the intensity maximum. Figure11.4 shows how a focused Gaussian light beam can produce the dipole gradient force,confining a collection of cold atoms near the beam axis. If the atoms have kinetic energyless than the potential arising from the dipole gradient force (Equation 11.101) it acts asan atom light trap.

–8 –6 –4 –2 0 2 4 6 8–80

–60

–40

–20

0

20

40

60

80

Beam Waist

Inte

nsity

Fdg

Figure 11.4 Gaussian light beam focused to a ‘beam waist’ radius. The intensity gradienttransverse to and along the beam axis acts as a confining force for two-level atoms when thelight beam is detuned far [(�ω)2 � (�/2)2]. Note that the dipole gradient force transverseto the beam is significantly greater than the force confining the atoms along thelongitudinal axis.


11.5.2 Radiation pressure force

Substituting the second term on the right of Equation 11.86 into Equation 11.85 resultsin an expression for the radiation pressure force:

Fr =12ε0E2

0kχ′′ (11.90)

and inserting Equation 11.76:

Fr =12ε0E2

0

[k|〈p12〉|23ε0h

(�/2

�ω2 +(�2

)2)]

(11.91)

The radiation pressure force peaks at zero detuning and is always in the direction oflight propagation k. The radiation force can be regarded as an impulse experienced byan atom when it absorbs or emits a quantum of light momentum hk, although classicallyit arises from the Lorentz force, F = v × B, where B is the magnetic induction field ofthe light wave.

11.5.3 Atom forces from the optical Bloch equations

So far we have handled spontaneous emission as a phenomenological loss term in Equa-tion 11.56. In fact, spontaneous emission results in a statistical distribution of final statesthat should be properly expressed in a set of coupled equations of the density matrix.Applied to atomic transitions, this density matrix results in the Optical Bloch Equations.A digression into the density matrix formalism would take us too far afield for the pur-poses of this chapter, so we will just state the pertinent results. Instead of an expressionfor the probability of finding the two-level system in state 2 as the square of the absolutevalue of the coefficient, |C2|2, we define the population of state 2 as

ρ22 =14 |�0|2

�ω2 +(�2

)2+ 1

2 |�0|2(11.92)

where γ is the same loss term we have been using and �0 is the same on-resonanceRabi frequency previously defined (Equation 11.36). Using ρ22 instead of |C2|2 leadsto a modification in the denominators of our expressions for the susceptibilities, χ ′,χ ′′,and consequently for the dipole gradient and radiation pressure forces. The modifiedexpressions are

Fdg =14ε0∇E0

2(r)

⎡⎣– |〈p12〉|2

3ε0h· �ω

�ω2 +((

�2

)2+ 1

2 |�0|2)⎤⎦ (11.93)


and

Fr =12ε0E2

0k

⎡⎣ |〈p12〉|2

3ε0h· �/2

�ω2 +((

�2

)2+ 1

2 |�0|2)⎤⎦ (11.94)

The force expressions retain their basic form: dispersive for the dipole gradient forceand absorptive for the radiation pressure force; but the denominators exhibit an extraterm that produces a broader effective width for each profile:

12|�0|2 = 1

2

(p12E0

h

)2

(11.95)

This term, one half the square of the Rabi frequency, is proportional to the electricfield intensity E2

0 and is often called ‘power broadening’ because the effective line pro-file broadens as the intensity of the applied light field increases. In fact, Equation 11.92shows that as the source intensity increases the fractional population of state 2 ap-proaches but does not exceed 1/2. The population of state 2 is said to ‘saturate’ at thatlimit. We can define a parameter S that indicates the degree of saturation as

S =12 |�0|2

�ω2 +(�2

)2 (11.96)

and rewrite the force expressions, Equations 11.93 and 11.94, in terms of the Rabifrequency and the saturation parameter:

Fdg = –h�ω6

∇S · 11 + S

(11.97)

and

Fr =hk3

(�

2

)· S1 + S

(11.98)

If we set S = 1 as the on-resonance tuning condition, �ω = 0, as a somewhat arbitrarydefinition of saturation, we have a saturation Rabi frequency:

�0sat =�√2

(11.99)

Figure 11.5 shows how S grows as the excitation power increases. The saturationradiation pressure force becomes

Fr = hk�

12(11.100)


0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Saturation Parameter S

Fr,

radi

atio

n pr

essu

re f

orce

(a.

u.)

Figure 11.5 Radiation pressure force Fr (arbitrary units) as a function of the saturationparameter S.

The dipole gradient force is conservative, and integrating over the spatial extentof the field gradient determines an atom-optical potential UT , where the subscriptindicates ‘trap’:

UT = –∫

Fdgdr =h�ω6

ln(1 + S) (11.101)

If the field is tuned relatively far from resonance so that �ω � � and S 1, then

UT � 112

�20

�ω(11.102)

When the optical trap (usually one or more focused laser beams) is tuned to the red ofresonance, UT presents a negative potential and the atom is drawn to the field ampli-tude maximum. By far the most frequent realisation of the optical trap is a single-mode,


Gaussian-profile, focused laser beam. The two-level atoms then find the potential min-imum along the laser beam axis. As we shall see shortly, atoms with sufficiently lowtemperature can be confined in such an optical trap. If the detuning is to the blue, thepotential becomes repulsive and the atoms are expelled from the laser beam centre. Bothattractive and repulsive potentials can be used to manipulate cold atoms.

11.5.4 Atom Doppler cooling

Consider the atom moving in the +z direction with velocity vz and counter-propagatingto the light wave detuned from resonance by �ωL, the effective detuning from the pointof view of the atom will be

�ω = �ωL + kvz (11.103)

In the rest frame of the atom the light appears shifted to the blue, δω = kvz, by theDoppler effect. The radiation pressure force acting on the atom as it scatters the lightwill be in the direction opposite to its motion. We have from Equations 11.94, 11.95, and11.103:

Fr± = ± hk6|�0|2

[�/2

(�ω ∓ kv)2 + (�2 )2 + 12 |�0|2

](11.104)

Suppose we have, as shown in Figure 11.6, two continuous wave Gaussian modes ±k,tuned within the atomic natural linewidth, propagating in the ±z directions, and wetake the amplitude of the net force to be the difference between the two componentsof the radiation pressure force when the atom is moving along z with vz. Figure 11.7shows that if �ω is tuned red ∼�/2, then the Doppler blue shift arising from the lightbeam propagating right to left in Figure 11.6 increases the scattering probability, therebyincreasing the opposing radiation pressure force. The light beam propagating left to rightis shifted further into the wing of the absorption profile. If kvz is small compared to �ω,we can expand the force expression in a Taylor expansion, and the net force acting onthe atom will be

z

vz

δω = kvz

ω0 + Δω ω0 + ΔωFigure 11.6 Schematic situation for 1-D atom-optical cooling. Two light beams tuned �ω to the redof the resonance frequency ω0 counter-propagate alongz and scatter from the atom moving along +z with vel-ocity vz. The Doppler light shift experienced by theatom of the light beam propagating from the right is+δω = kvz; from the left, –δω.


–50 –40 –30 –20 –10 0 10 20 30 40 500

0.005

0.01

0.015

0.02

0.025

Fr (

Δω)

Δω

Δω

δω

ωL

δω

Figure 11.7 Absorption profile for the atom moving with velocity +vz in the presence of twocounter-propagating light beams tuned ωL = ω0 –�ω � –�/2. The blue Doppler shift +δω (verticalline right of ωL ) shifts absorption closer to line centre. The red Doppler shift –δω (shortest vertical lineto the left of ωL ) shifts absorption further into the absorption wing.

Fnet � hk6|�0|2

(�

2

)⎡⎢⎣ 4kvz�ω(�ω2 +

(�2

)2+ 1

2 |�0|2)2⎤⎥⎦ (11.105)

This expression can be simplified somewhat by setting �ω � �/2, which in practicalexperiments is close to reality, and defining an on-resonant saturation parameter as

S0 =|�0|2�2/2

(11.106)

so that

Fnet � hk3S0

(1�

)[8kvz�ω

(2 + S0)2

](11.107)


Furthermore, in practical cases atoms subject to this radiation pressure force are spin-polarised so that we can drop the averaging over the transition dipole orientation:

Fnet � hk(8kvz�ω�

)[S0

(2 + S0)2

](11.108)

We see that when the detuning is negative (red detuned), the Doppler-modified radiationpressure force acts in a direction opposite to the atom propagation and is proportionalto the atomic velocity. Figure 11.8 shows how, within a relatively narrow range of atommotion kv, the radiation pressure force acts in a direction opposite to the motion, whileFigure 11.7 shows this restoring force also increases the dissipative light scattering. Theatom undergoes an oscillatory motion along ±z subject to a velocity-dependent restor-ing force. This situation corresponds to the damped harmonic oscillator expressed byEquation 11.13. The damping constant β is given by

β = 8hk2�ω

�

S0(2 + S0)2

(11.109)

–10 –8 –6 –4 –2 0 2 4 6 8 10–0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

0.4

kvz (a.u.)

Fr (

a.u.

)

Figure 11.8 Restoring force plotted against kvz. Note the approximate linear dependenceof Fr when kvz does not deviate too far from zero such that the Doppler shift remainswithin an interval � �/2. This linear range is indicated by the vertical dashed lines.


Assuming a typical red detuning of �/2, we have

β = 4hk2S0

(2 + S0)2(11.110)

and further assuming that the transition is far from saturation,

β � hk2S0 (11.111)

The time to damp the atom motion of m is

τ =m2β

(11.112)

The frictional force removes kinetic energy from the atom and dissipates it as spontan-eous emission. The atom can be said to be ‘cooled’ by this dissipative light force, thecooling rate being the inverse of the damping time:

ρcool � 2hk2

mS0 (11.113)

The atom does not cool indefinitely however. The cooling rate must be balanced by theheating rate arising from the recoil impulsion each time the atom absorbs a photon. Thisheating rate is related to the fluctuation coefficient of the radiation pressure force2, D ,

D = 3h2k2�S0 (11.114)

When the heating and cooling rates are balanced, we find the effective steady-statetemperature for the atom:

Teff =D

3kBmρcool(11.115)

Teff =12h�kB

(11.116)

where kB is the Boltzmann constant. This effective temperature is called the ‘Dopplercooling limit’ and corresponds to a temperature of several hundred micro-Kelvin foralkali atoms such as sodium or rubidium. In fact, because these atoms are not truetwo-level atoms, other cooling mechanisms involving the hyperfine levels can drive thesteady-state temperature much lower than the Doppler cooling limit. Discussion of thesesub-Doppler cooling mechanisms is, however, beyond the scope of this chapter.

2 The fluctuation force is discussed at length in the thesis of J. Dalibard, referenced in Further reading inSection 11.8.


11.6 Summary

The chapter begins with the atom as a damped harmonic oscillator. The next step isto consider the more realistic model of the semiclassical two-level atom where we findthe transition rate between levels and from which we can calculate the Einstein B12 co-efficient, the spontaneous emission loss, and develop expressions from the polarisationand susceptibilities (real and imaginary parts). Light-atom forces (dipole gradient andradiation pressure) are then expressed in terms of the susceptibilities and field or inten-sity gradients (dipole-gradient force) or phase gradient (radiation pressure force). Thequestion is again examined from the perspective of the optical Bloch equations and thesaturation parameter S. The chapter ends with a discussion of the simplest atom-opticalcooling process–Doppler cooling.

11.7 Exercises

1. Given that the spontaneous emission lifetime τ = 1/� of the sodium atomNa(3p) 2P3/2 level is 16 ns, calculate the magnitude of the transition dipole momentof the Na(3s) 2S1/2→ Na(3p) 2P3/2 transition.

2. The wavelength of the resonant Na(3s) 2S1/2 → Na(3p) 2P3/2 transition in atomicsodium is 589 nm. Calculate the transverse dipole gradient force on a Naatom illuminated by a Gaussian mode laser focused to a beam waist of 1μmand detuned to the red of the resonance frequency by 10�. Take the lightsource to be a single-mode, continuous-wave laser with an average power of100mW.

3. Calculate the trap depth transverse potential energy well (in units of kBT) forExercise 2. What is the maximum atom temperature that would still permit atomconfinement?

4. Calculate the restoring force on a Na atom moving in the +z direction with velocityv = 10ms–1. Assume a light detuning �ω = –�/2 and a saturation parameter S0 = 1.

5. Calculate the Doppler cooling limiting temperature for a Na atom and the timerequired to cool the atom from ambient temperature to this limit.


1. J.-P Pérez, R. Carles, and R. Fleckinger, Électromagnétism, Fondements et Applications,3ème édition, Masson (1997).

2. C. Cohen-Tannoudji, B. Diu, and F. Laloé, Quantum Mechanics, Wiley (1977).

Further reading 327

3. J. Weiner, P.-T. Ho, Light-Matter Interaction: Fundamentals and Applications, Wiley-Interscience (2003).

4. John Weiner, Cold and Ultracold Collisions in Quantum Microscopic and MesoscopicSystems, Cambridge (2003).

5. Jean Dalibard, Le Role des Fluctuations dans la Dynamique d’un Atome Couplé auChamp Électromagnétique, Thèse de Doctorat d’Etat, Université Pierre et Marie Curie(1986).

12

Radiation in Classical and QuantalAtoms

12.1 Introduction

The hydrogen atom is the simplest atomic structure interacting with light in the opticalregime. We know from the earliest days of atomic spectroscopy that the hydrogen atomemits and absorbs dipole radiation. But the emitted and radiated light only appears atcertain frequencies; the hydrogen atom does not exhibit a continuous spectrum as theclassical dipole antenna. The reason is that the atom is a quantum object, not a classicaldipole. Nevertheless, a ‘semiclassical’ picture of the light–atom interaction is possible. Inthis picture, the electron, with dynamics defined by the quantal nature of atomic internalstates, still acts as a receiving antenna or a source of classical (usually dipole) radiation.This hybrid picture of quantal matter and classical electrodynamics captures a greatdeal of the atom-optical physics. Non-relativistic quantum mechanics and the classicalradiation field combine to explain almost all the features of the hydrogen atom spectrum.We will use these tools to develop an explanation for the quantisation of the energy levelsof the atom and the ‘selection rules’ for dipole transitions between them.

12.2 Dipole emission of an atomic electron

Here for convenience we summarise some of the discussion in Section 11.2.In Section 2.8 we considered a classical electric dipole aligned along the z-axis and

wrote the dipole as

p(t) = q0a cosωtz

If we consider an atom as an electron bound to a massive proton, subject to a radialrestoring force centred at the proton position, then we can write down a similar dipoleexpression,

p(t) = –er(t) p0 = –er0 (12.1)


Dipole emission of an atomic electron 329

where –e is the electron charge, r is the radial distance from the atom centre, and r0 isthe maximum radial distance. The electron is pictured as vibrating radially through theproton position and in this simple model exhibits no angular momentum from orbitalmotion about the charge centre. Using the result for power emitted by a classical dipole,Equation 2.157, we can write the optical-cycle-averaged power emitted by the atom as

〈W 〉 = 14πε0

ω40p

20

3c3=

e2

4πε0

ω40r

20

3c3(12.2)

For harmonic oscillation the maximum acceleration is a0 = –ω20r0 and

〈W 〉 = e2

4πε0

a203c3

(12.3)

Writing the cycle-averaged acceleration as 〈a · a〉 = ⟨a2⟩ = (1/2)a20, we have

〈W 〉 = e2

4πε0

2⟨a2⟩

3c3(12.4)

and we see that the average power emitted by the oscillating electron is proportional tothe average square of the electron acceleration. A charged particle must be acceleratingto emit radiation; an electron moving at constant velocity, not subject to external forces,does not emit.

The energy (cycle-averaged) emitted by the oscillator must come from mechanical(kinetic + potential) energy, Emech:

Emech =12meω

20r

20 (12.5)

where me is the electron mass. The average power emitted by the oscillator can beequated to the diminution of mechanical energy with time:

〈W 〉 = –Emech

τ(12.6)

From Equations 12.2 and 12.5, and 12.6 we see that

τ =(

e2

4πε0

)–1 3mec3

2ω20

(12.7)

The ‘lifetime’ of the oscillator τ can be written a little more simply by introducing theclassical radius of the electron,

rc =e2

4πε0

1mec2

� 2.8× 10–15 m (12.8)

330 Radiation in Classical and Quantal Atoms

and

τ =3c

2rcω20

(12.9)

In differential form Equation 12.6 is written as

〈W 〉 = –dEmech

dt

so

dEmech

Emech= –

dtτ

(12.10)

and

Emech(t) = Emech(0)e–t/τ (12.11)

We see that the internal energy of the atom, modelled as a classical electron oscillat-ing around a positive charge centre, decreases exponentially with time as the energy isradiated away.

A specific example provides some feeling for the relevant time and energy scales.Consider a sodium atom, structured with 10 electrons in a ‘closed shell’ and one electronoutside. This exposed or ‘active’ electron is subject to an effective positive charge at thenucleus roughly equal to that of one proton. The potential and electric field experiencedby the active electron is radial and the simple dipole oscillator model might therefore beappropriate. Emission from the first excited state to the ground state is concentrated in anarrow band of wavelengths centred at 590 nm, the well-known ‘D line’ of atomic spec-troscopy. Considering this emission as emanating from a dipole oscillator with resonantfrequency ω0,

ω0 =2πcλ

= 3.2× 1015 s–1

and

τ =3c

2rcω20

� 16× 10–9 s

This result is actually quite close to the measured lifetime of the first excited state ofthe sodium atom and lends credence to the harmonic oscillator model. In reality, how-ever, the atom must be treated quantum mechanically because there is no lower limitto the classical radiative emission, and the classical electron would fall into the nucleusas its mechanical energy diminishes to zero. Furthermore, the first excited state with

Radiative damping and electron scattering 331

a quantum-mechanically ‘allowed’ transition possesses one quantum of orbital angularmomentum, and this angular momentum does not appear in the simple oscillator model.

12.3 Radiative damping and electron scattering

Nevertheless, the damped harmonic oscillator model provides some insight into thephysics of atomic emission, absorption of radiation, and atomic scattering. We canwrite down the equation of motion of the classical, radiatively damped, externally drivenharmonic oscillator, moving along the z-axis, as

d2zdt2

+1τ

dzdt

+ ω20z = –

eEzme

e–iωt (12.12)

where as before ω0 is the characteristic frequency of the oscillator, e,me are the electroncharge and mass, respectively, τ the classical oscillator lifetime, and Ez is the amplitudeof the external driving field oscillating at frequency ω. The solution to this equation ofmotion is

z =–eEze–iωt

me[(ω20 – ω

2)– iω/τ

] (12.13)

From this result we can write down the frequency-dependent dipole moment,

p = –ezz =–eEze–iωt

me[(ω20 – ω

2)– iω/τ

] z (12.14)

From Equation 12.2 we have for the average dissipated power,

〈W 〉 = e4E2zω

4

12πε0m2e

[(ω20 – ω

2)2

+ (ω/τ)2] (12.15)

Using the classical electron radius, Equation 11.9, we can write this expression for thedissipated power as the product of an energy flux and a scattering cross section,

〈W 〉 =(ε0E2

z

2

)c︸︷︷︸

energy flux

·{[

8πr2c3

][ω4(

ω20 – ω

2)2

+ (ω/τ)2

]}︸︷︷︸

cross section

(12.16)

where the scattering cross section is defined as

σ (ω) =[8πr2c3

][ω4(

ω20 – ω

2)2 + (ω/τ)2

](12.17)


12.4 The Schrödinger equation for the hydrogen atom

The basic expression for the Schrödinger equation of any atomic system is described byan operator equation,

H�n(r, t) = ih∂�n(r, t)

∂t(12.18)

where �n(r, t) represents the state n of the atom, and H is the Hamiltonian operator,H = ih∂ /∂t. According to the postulates of quantum mechanics, a characteristic or‘eigen’ state of a system must be represented by a wave function that is single-valued,normalised, and orthogonal to all the other eigenstates of the system. An arbitrary systemstate may be expressed as a linear combination of these eigenstates that form a ‘completeset’ in a linear, orthogonal vector space. If the state is ‘stationary’ or time independent,as it is for the isolated atom, we may write the state function as a product of a spatialterm ψ(r) and a time-varying phase, exp(–iωt):

�(r, t) = ψ(r)e–iωt (12.19)

The frequency of the phase is related to the energy En of the stationary state bythe Planck relation, En = hω. The spatial, time-independent part can factor, and beseparated by, substitution of Equation 12.19 into Equation 12.18 with the result

H [ψ(r) – En] = 0 (12.20)

where ψ(r) is the spatial wave function that is the solution to the time-independentSchrödinger equation, Equation 12.20. Now the Hamiltonian operator H is constructedfrom the Bohr correspondence principle that permits us to identify certain mathematicaloperators with their dynamical variable counterparts in the Hamiltonian formulation ofclassical mechanics. The classical Hamiltonian H expresses the energy of a system. Forexample,

H = T + V (12.21)

where T is the kinetic energy and V the potential energy. The correspondence prin-ciple is a rule that indicates the equivalence of these classical functions to the operatorsrepresenting them in quantum mechanics. For example,

H → H T → T V → V (12.22)

The kinetic and potential energies themselves are functions of position and momentum,which correspond to the operators r and p, respectively. The ‘operator’ for the positionis the position itself,

r→ r (12.23)

The Schrödinger equation for the hydrogen atom 333

and the operator for the momentum is

p→ –ih∇ (12.24)

Therefore the operator expression for the kinetic energy of a particle with mass mbecomes

T =p2

2m→ T = –

h2

2m∇2 (12.25)

and if the potential energy is a function only of position, then

V (r)→ V (r) (12.26)

From Equations 12.25 and 12.26 we can construct the Hamiltonian operator and theSchrödinger operator equation for the energy of the system,

[H – En

]ψ = 0→

[–h2

2m∇2 + V (r) – En

]ψ = 0 (12.27)

The physically allowed solutions to the time-independent Schrödinger equation, Equa-tion 12.27, constitute the ‘states’ of the physical system described by the energy terms ofthe Hamiltonian. In the case of a hydrogen atom, the system is quite simple, consistingof a single proton and a single electron bound together by the electrostatic Coulombpotential. Strictly speaking, this situation is a two-particle problem, and the kinetic en-ergy term in Equation 12.27 should be the sum of two terms for the kinetic energies ofthe proton and electron. However, the rest mass of the proton is greater than that of theelectron by a factor of nearly 2000, and it is convenient to consider the ‘reduced mass’μ of the equivalent single-particle problem:

μ =memp

me +mp� me (12.28)

and the centre-of-mass (cm) coordinate,

rcm =mprp +meremp +me

� rp (12.29)

So to a very good approximation we can consider a model system where the reducedmass can, essentially, be considered that of the electron and the centre-of-mass coord-inate considered that of the proton. By placing the origin of the coordinate system atthe centre-of-mass position we simplify the problem to a single electron moving about astationary positive charge centred at the origin. For all practical purposes, therefore, wecan write the time-independent Schrödinger equation for the hydrogen atom as


–h2

2me∇2ψ(x, y, z) –

14πε0

e2(x2 + y2 + z2

)1/2ψ(x, y, z) – Enψ(x, y, z) = 0 (12.30)

where the Laplacian operator ∇2 in Cartesian coordinates is

∇2 =∂2

∂x2+∂2

∂y2+∂2

∂z2(12.31)

and the Coulomb potential is

V = –1

4πε0

e2(x2 + y2 + z2

)1/2 (12.32)

Since the Coulomb potential term exhibits spherical symmetry, Equation 12.30 can besolved far more easily in spherical coordinates, r, θ ,φ, where,

x = r sin θ cosϕ (12.33)

y = r sin θ sinϕ (12.34)

z = r cos θ (12.35)

r =(x2 + y2 + z2

)1/2(12.36)

and the Laplacian operator ∇2 is expressed as

∇2 =1r2∂

∂r

(r2∂

∂r

)+

1

r2 sin2 θ

∂2

∂ϕ2+

1r2 sin θ

∂

∂θ

(sin θ

∂

∂θ

)(12.37)

The Schrödinger equation in spherical coordinates takes the form

–h2

2me∇2ψ(r, θ ,ϕ) –

14πε0

e2

rψ(r, θ ,ϕ) – Enψ(r, θ ,ϕ) = 0 (12.38)

The relation between Cartesian and spherical coordinates and the representation of theLaplacian operator in spherical coordinates is discussed in AppendixD. In sphericalcoordinates the wave function solution to Equation 12.30 factors into three functions ofeach coordinate,

ψ(r, θ ,ϕ) = R(r)P(θ)(ϕ) (12.39)


Substituting this form back into Equation 12.38 results in the separation into threedifferential equations in r, θ ,ϕ, respectively:

1r2ddr

(r2dRnldr

)+

2μ

h2[E – V (r)]Rnl = l(l + 1)

Rnlr2

(12.40)

–1

sin θddθ

(sin θ

dPldθ

)+

Plsin2 θ

=l(l + 1)

m2l

Pl (12.41)

d2dϕ2

= –m2l (12.42)

The expressions l(l + 1) and – (ml)2 are constants of separation.

12.4.1 Radial equation

The independent variable of the radial equation, Equation 12.40, is r, the distance of theelectron from the origin. The physically admissible solutions are labelled by the quantumnumber integers n, l. By ‘physically admissible’ we mean solutions so that Rnl → 0 asr→∞ and Rnl remains finite as r→ 0. There are in fact a family of admissible solutions,each member of which is identified by a unique n, l label. The n quantum number iscalled the principal quantum number and is directly identified with the total energy ofthe atom:

En =μe4

(4πε0)2 2h2n2� –

mee4

(4πε0)2 2h2n2(12.43)

where n can take on the integer values n = l, l + 1, l + 2, . . . and l can be an integerl = 0, 1, 2, . . .. The radial solutions take the form of three factors: a normalisation fac-tor, a polynomial, and an exponential. In fact, Equation 12.40 can be rearranged intoa form that bears a striking resemblance to the differential equation whose solutionsare the ‘modified Laguerre functions’. The rearranged form is obtained by multiplyingEquation 12.40 by –2μ/h2 and transposing two terms:

–h2

2μ1r2ddr

(r2dRnldr

)–

e2

4πε0rRnl +

h2

2μl(l + 1)r2

Rnl = EnRnl (12.44)

The differential equation for the modified Laguerre functions and its solutions, Equa-tions F.18 and F.19, are discussed in Appendix F. Now if we make the followingsubstitutions, Equation 12.44 can be cast into the form of EquationF.19:

ρ = αr and α =

√–8μE

h2(12.45)


where it is understood that E < 0 since we only consider the allowed bound-state ener-gies of the hydrogen atom, and the reference zero of energy is at the ionisation limitr → ∞. The quantity α has units of m–1 so ρ is unitless. We make an additionalsubstitution to consolidate many constants,

λ =μe2

2πε0αh2(12.46)

In order for the λ substitution to be consistent with the physically admissible values of theenergy En, λ must be restricted to integers, λ = 1, 2, 3, . . ., and defining χ(ρ) = R(ρ/α),we can recast Equation 12.44 as,

1ρ2

ddρ

(ρ2dχ(ρ)dρ

)+(λ

ρ–14–l(l + 1)ρ2

)χ(ρ) = 0 (12.47)

Using one last substitution trick,

1ρ2

ddρρ2dχdρ

=1ρ

d2

dρ2(ρχ) (12.48)

we have, finally,

d2(ρχ)dρ2

+(–14+λ

ρ–l(l + 1)ρ2

)ρχ = 0 (12.49)

Now we compare this last result to EquationF.19 rewritten here for convenience,

d2kn(x)

dx2+(–14+

2n + k + 12x

–k2 – 14x2

)kn(x) = 0

We see that Equation 12.49 becomes equivalent to EquationF.19 if

ρ ↔ x l ↔ k – 12

λ→ n + l + 1 (12.50)

Then the solutions to Equation 12.49 are

ρχ(ρ) = e–ρ/2ρ l+1L2l+1λ–l–1(ρ) (12.51)

equivalent to the solutions for EquationF.19:

kn(x) = e–x/2x(k+1)/2Lkn(x) (12.52)

where Lkn(x) are the associated Laguerre polynomials. Thus, Equation 12.51 expressesthe solutions to Equation 12.49 that in turn is just the hydrogen radial equation, Equa-tion 12.40, in a different guise. Now we recover from the scaling parameters α and ρ


terms of physically meaningful quantities. From the expression for the allowed boundenergies En of the hydrogen atom, Equation 12.43, we can write α and ρ in terms of nthe principal quantum number:

α =2na0

and ρ =2na0

r (12.53)

where a0 = 4πε0h2/mee2, the well-known atomic radius of the Bohr model. Finally, wecan write out the solutions to the hydrogen radial equation, Equation 12.40,

Rnl(r) = (αr)lL2l+1n–l–1(αr)e

–αr/2 (12.54)

In quantum mechanics the solutions to the Schrödinger equation play an analogous roleto force fields in the Maxwell equations. Although the wave function is not a vectorfunction, it is, in general, complex. Just as in electrodynamics, the force field F itselfis not observed but the intensity, FF∗, guaranteed to be a real quantity, is the detectedsignal, so in quantum mechanics it is the probability density, ψψ∗ that connects theoryto experiment. Note that EE∗ and HH∗ are directly proportional to the energy densityof an electromagnetic field. It is therefore plausible to assign ψψ∗ the role of probabilitydensity. When ψ is a function only of space and is the solution to a single-particle Schrö-dinger equation, then ψ(r)ψ∗(r) is interpreted as the probability density of finding theparticle at position r. Evidently then,∫

all spaceψ(r)ψ∗(r) dr = 1 (12.55)

and when this condition is fulfilled, the wave function is said to be normalised. In orderto be consistent with this probability density interpretation, the functions Rnl(r) must beprefixed by a constant normalising factor

√NRnl such that

1NRnl

∫ ∞0

R∗nl(r)Rnl(r)r2 dr = 1 (12.56)

It can be shown from the normalisation condition of the associated Laguerre functions,EquationF.16 in Appendix F, that

NRnl =(na0

2

)3 2n(n + l)!(n – l – 1)!

(12.57)

so that, finally, the normalised radial solutions to the hydrogen atom Schrödingerequation are:

Rnl(r) =

[(2na0

)3 (n – l – 1)!2n(n + l)!

]1/2 (2na0

r)lL2l+1n–l–1e

–(2/na0)r/2 (12.58)


Table 12.1 Hydrogen radial wave functions Rnl , n=1 – 3;l=0

n l Rnl

1 0(

1a0

)3/22e–r/a0

2 0(

12a0

)3/2 [2 – r

a0

]e–r/2a0

3 0(

13a0

)3/22[1 – 2

(ra0

)+ 2

27

(ra0

)2]e–r/3a0

00

0.1

0.2

0.3

P(r

)

0.4

0.5 (n=1, ℓ=0)

(n=2, ℓ=0)

(n=3, ℓ=0)

0.6

5 10 15 20 25 30r/a0

Figure 12.1 Normalised radial probability density as a functionof the radial coordinate (in units of a0) for three radial wavefunctions.

Figure 12.1 shows the normalised radial probability density functions,

P(r) = R∗nlRnl r2 (12.59)

for the hydrogen atom ground state n = 1, l = 0 and two excited states. Table 12.1 liststhe radial functions, Rnl used to plot the probability densities of Figure 12.1.

12.4.2 Angular equations

12.4.2.1 Functions of the azimuthal angle ϕ

The solutions to Equation 12.41 are a family of functions of the polar angle θ and areusually denoted Pl(cos θ) with cos θ as the independent variable. This family is calledthe associated Legendre functions, and their properties are discussed in Appendix G.The solutions to Equation 12.42, the functions of the azimuthal angle ϕ, can be obtainedby inspection. It is easy to verify that

ml (ϕ) = eimlϕ (12.60)


According to the postulates of quantummechanics, the solution to a differential equationmust be single-valued in the independent variable in order for the function to suit-ably represent a physical state. Therefore, as ϕ cycles through integer multiples of 2π ,increasing from 0 → 2π , 2π → 4π , . . . the value of the function ml should remainsingle-valued. Thus, ml (ϕ) must have the same value as ml (ϕ

′) if ϕ′ = ϕ + n2π wheren = 1, 2, 3, . . .. In order to assure this behaviour, the value of ml must be an integer,ml = ±1,±2,±3, . . .. Furthermore, ml is subject to a normalisation requirementanalogous to that of Equation 12.56:

1N

∫ 2π

0∗ml (ϕ)ml (ϕ) dϕ = 1 (12.61)

The normalised azimuthal wave function is therefore

ml (ϕ) =1√2π

ei(±ml)ϕ (12.62)

12.4.2.2 Functions of the polar angle θ

The solution to Equation 12.41 constitutes a family of functions, Pmll (cos θ) called theassociated Legendre functions. The properties of the Legendre functions and the Le-gendre polynomials, the relevant functions when ml = 0, are discussed in some detailin Appendix G. These functions describe the polar-angle dependence of the hydro-gen atom wave function, ψ(r, θ ,ϕ) = RnlPmll (cos θ)ml(ϕ). Just as Rnl and ml must benormalised, so must the functions Pmll (cos θ). The normalisation requirement specifiesthat

1N

∫ π

0P∗mll (cos θ)Pmll (cos θ) sin θ dθ = 1 (12.63)

with N expressed by

N =2

2l + 1· (l +ml)!(l –ml)!

(12.64)

The family of functions that are the product of Pl(cos θ) and (ϕ) is called the sphericalharmonics, Yml

l (θ ,ϕ). The normalised spherical harmonics are

Ymll (θ ,ϕ) = (–1)ml

√2l + 14π

· (l –ml)!(l +ml)!

Pmll (cos θ)eimlϕ (12.65)

and ∫ π

0

∫ 2π

0Y ∗mll (θ ,ϕ)Yml

l (θ ,ϕ) sin θ dθ dϕ = 1 (12.66)


In order to maintain physically admissible, well-behaved functions there are certainrestrictions on the function labels, n, l,ml . We summarise them here without proof:

n = 1, 2, 3, . . . (12.67)

l = 0, 1, 2, . . . n – 1 (12.68)

ml = –l, –l + 1, –l + 2, . . . 0 . . . , l – 2, l – 1, l (12.69)

Table 12.2 lists the spherical harmonic functions in spherical coordinates up to n = 3.Putting the spherical harmonic angular wave functions together with the radial wave

functions gives us, finally, the full three-dimensional wave functions that specify thatquantum state of the hydrogen atom:

ψ(r, θ ,ϕ)nlml = Rnl(r)Ylml (θ ,ϕ) (12.70)

The expression ψ∗nlml (r, θϕ)ψnlml (r, θ ,ϕ) is the probability density of finding the electronat r.

r = rr polar coordinates (12.71)

r = xx + yy + zz Cartesian coordinates (12.72)

The probability dP of finding the electron in a volume interval between V and V +dV is

dP = ψ∗nlml (r, θϕ)ψnlml (r, θ ,ϕ) dV (12.73)

dP = ψ∗nlml (r, θϕ)ψnlml (r, θ ,ϕ) r2 sin θ dθ dϕ dr (12.74)

Table 12.2 Angular functions, spherical harmonics Ymll .

n l ml Ymll

1 0 0 12

√1π

2 1 –1 12

√32π sin θe–ϕ

2 1 0 12

√32π cos θ

2 1 1 –12

√32π sin θeϕ

3 2 –2 14

√152π sin2 θe–2ϕ

3 2 –1 12

√152π sin θ cos θe–ϕ

3 2 0 14

√5π

(cos2 θ – 1

)3 2 1 –12

√152π sin θ cos θeiϕ

3 2 2 14

√152π sin2 θei2ϕ


Table 12.3 Hydrogen wave functions RnlYmll , n=1 – 3.

n l ml Rnl Ymll

1 0 0 1√π

(1a0

)3/2e–r/a0 1

2 0 0 14√2π

(1a0

)3/2 [2 –

(ra0

)]e–r/2a0 1

2 1 0 14√2π

(1a0

)3/2 ( ra0

)e–r/2a0 cos θ

2 1 ±1 18√π

(1a0

)3/2 ( ra0

)e–r/2a0 sin θe±iϕ

3 0 0 181√3π

(1a0

)3/2 [27 – 18

(ra0

)+ 2

(ra0

)2]e–r/3a0 1

3 1 0 181

√2π

(1a0

)3/2 (6 – r

a0

) (ra0

)e–r/3a0 cos θ

3 1 ±1 181√π

(1a0

)3/2 (6 – r

a0

) (ra0

)e–r/3a0 sin θe±iϕ

3 2 0 181√6π

(1a0

)3/2 ( r2a0

)2e–r/3a0 3 cos2 θ – 1

3 2 ±1 181√π

(1a0

)3/2 ( r2a0

)2e–r/3a0 sin θ cos θe±iϕ

3 2 ±2 1162√π

(1a0

)3/2 ( r2a0

)2e–r/3a0 sin2 θe±i2ϕ

and ∫all space

dP =∫ ∞0

∫ π

0

∫ 2π

0ψ∗nlmlψnlml r

2 sin θ dθ dϕ dr = 1 (12.75)

Table 12.3 lists the normalised ψnlml solutions to the hydrogen atom Schrödingerequation for n = 1 – 3.

12.4.3 Orthogonality

If ψn,l,ml represents an eigenstate of the hydrogen atom with an eigen energy and angu-lar momentum, then according to the postulates of quantum mechanics it must be notonly single-valued and normalised, but also orthogonal to all other states with differentenergies or angular momenta. More succinctly,∫

all space

ψ∗n,l,mlψn′ ,l′,m′l dτ = δnn′δll ′δmlm′l (12.76)

where the δ function takes the value of unity when n, l,ml are equal to their primedcounterparts but is null otherwise. Wave functions that exhibit this property are said tobe ‘orthonormal’ because they are orthogonal and normalised. In polar coordinates thewave functions of the hydrogen atom are products of radial and angular functions,


ψn,l,ml (r, θ ,ϕ) = Rnl(r)Ylml (θ ,ϕ) = Rnl(r)Pmll (cos θ)ml (ϕ) (12.77)

and therefore these radial and angular functions must be orthonormal as well.

12.5 State energy and angular momentum

Now that we have the functional form of the radial and angular solutions to the Schrö-dinger equation for the hydrogen atom, we can examine their properties in moredetail.

12.5.1 Eigen energies

Considering the time-independent Schrödinger equation as an operator equation, Equa-tion 12.20, we see that an eigenstate, when operated upon by the energy operator H ,returns the energy eigenvalue of that state,

Hψn,l,ml = Enψn,l,ml (12.78)

The eigen energies En are labelled by the principal quantum number n and are char-acterised by the analytical expression given in Equation 12.43. The constants in thisexpression can be collected together to facilitate numerical calculation in convenientunits:

En = –2.18× 10–18

n2J = –

13.6n2

eV = –1.097× 105

n2cm–1 (12.79)

The allowed values of n are n = 1, 2, 3, . . .. The lowest lying (ground) state of the atomcorresponds to n = 1, and E1 = –13.6 eV. The first excited state corresponds to n = 2,and E2 = –3.40 eV, and so forth. As n → ∞, En → 0, and the zero-reference energycorresponds to the ionisation limit where the electron is no longer bound by the Coulombpotential. Figure 12.2 shows a diagram of the hydrogen atom energy levels bound withinthe Coulomb potential.

12.5.2 Angular momentum

In addition to energy, the states of the hydrogen atom may exhibit orbital angular mo-mentum. In the classical Bohr model, the electron rotated around the proton in variousorbital configurations reminiscent of the planets around a star. The uncertainty prin-ciple no longer permits us to make such a literal interpretation, but the Schrödingerequation does show us how to calculate the orbital angular momentum of the hydro-gen atom states. We can obtain the quantum mechanical operator corresponding tothe classical angular momentum by applying the ‘correspondence principle’ discussedin Section 12.4. From classical mechanics we have, for the angular momentum of the

State energy and angular momentum 343

0

n=1

n=2V(r)n=3

Hydrogen Bound States

n=4

1 2 3 r

Figure 12.2 Coulomb potential to which the bound electron inthe hydrogen atom is subject. The lowest four bound states(n = 1 – 4) are shown.

electron L, from the position r, and the linear momentum p,

L = r× p (12.80)

and the three Cartesian components:

Lx = ypz – zpy (12.81)

Ly = zpx – xpz (12.82)

Lz = xpy – ypx (12.83)

Then from the correspondence principle,

r→ r (12.84)

p→ –ih∇ (12.85)

The operator equivalents of three angular momentum components are therefore

Lx = –ih(y∂

∂z– z

∂

∂y

)(12.86)

Ly = –ih(z∂

∂x– x

∂

∂z

)(12.87)

Lz = –ih(x∂

∂y– y

∂

∂x

)(12.88)


and the operator corresponding to the square of the total orbital angular momentum is

L2 =(L2x + L2

y + L2z

)(12.89)

In polar coordinates, as discussed in AppendixD,we have

Lx = ih[sinϕ

∂

∂θ+(

1tan θ

)cosϕ

∂

∂ϕ

](12.90)

Ly = ih[– cosϕ

∂

∂θ+(

1tan θ

)sinϕ

∂

∂ϕ

](12.91)

Lz = –ih∂

∂ϕ(12.92)

and Equation 12.89 transforms to

L2 = –h2[

1sin θ

∂

∂θ

(sin θ

∂

∂θ

)+

1

sin2 θ

∂2

∂ϕ2

](12.93)

From Equations 12.41 and 12.42 and the definition of the spherical harmonic functions,Equation 12.65, we see that

L2P0l (θ) = h2l(l + 1)P0

l (θ) (12.94)

LzYl,ml (θ ,ϕ) = hmlYl,ml (θ ,ϕ) (12.95)

The value of the orbital angular momentum for a given eigenstate of the hydrogen atomis therefore:

L = h√l(l + 1) (12.96)

and the Lz component is given by

Lz = hml (12.97)

Thus, we see that the ground state exhibits no orbital angular momentum, with n = 1,l = 0, and L,Lz = 0. The first excited state, n = 2, can have values of l = 0, 1. Therefore,the n = 2 level exhibits two states of orbital angular momentum, L = 0 and L = h

√2.

The level with l = 1 possesses three possible values of Lz: h, 0, –h. It is already clearfrom these results that the Schrödinger equation yields a physical interpretation of theinternal atomic structure quite different from that of the Bohr orbits. Firstly, since theground state has no angular momentum, we cannot imagine the electron as orbitingthe nucleus. The electron is restricted to radial motion. Secondly, even when the orbitalstate does have non-zero angular momentum, the projection L on the z-axis can onlytake on discrete values. In the case of l = 1, only three Lz values are allowed. In aclassical orbit, there would be no restriction on Lz from –L to L. A third nonclassical

Real orbitals 345

feature of the angular momentum is the peculiar status of the two other components,Lx,Ly. A consequence of the uncertainty principle is that the position and momentumof a particle cannot be known (measured) simultaneously. If all three components of theangular momentumwere measurable and the magnitude of L was known as well, then theposition and momentum of the electron in the atom would be determined simultaneouslyin violation of this fundamental postulate of the quantum theory.

12.6 Real orbitals

It is sometimes useful to consider linear combinations of the spherical harmonics, whichof course, are themselves solutions to the angular part of the Schrödinger equation.In particular, linear combinations can be chosen so that resulting solutions are real.These real one-electron functions are termed ‘orbitals’ and they play an important rolein molecular structure. These real orbitals give spatial directionality to the angular wavefunctions and can help to interpret the geometry of chemical bonding when atoms com-bine to form molecules. Linear combinations of l = 1 solutions to the Schrödingerequation are called p-orbitals. They exhibit electron probability density localised alongthe three orthogonal x, y, z coordinate axes, as shown in the first row of Figure 12.3.Combinations of the five l = 2 solutions result in d-orbitals. The electron probabilitydensity of three of these orbitals are indicated in the second row of Figure 12.3. Table12.4 lists the linear combinations of spherical harmonics for l = 1 and l = 2. The labels

Figure 12.3 Plots of the p-orbitals and three of the d-oribitalsshowing their spatial directions and orientations.


Table 12.4 Real orbitals from linear combinations of spherical harmonics.

n l real orbital linear combination Cartesian coordinates

2 1 px√

12

(Y–11 – Y1

1

) √34π · xr

2 1 py i√

12

(Y–11 +Y1

1

) √34π · yr

2 1 pz Y01

√34π · zr

3 2 dx2–y2√

12

(Y–22 +Y2

2

)14

√15π · x

2–y2

r2

3 2 dxy i√

12

(Y–22 –Y2

2

)12

√15π · xyr2

3 2 dxz√

12

(Y–12 – Y1

2

)12

√15π · xzr2

3 2 dyz i√

12

(Y–12 +Y1

2

)12

√15π · yzr2

3 2 dz2 Y02

14

√5π · 2z

2–x2–y2

r2

for the real orbitals arise from the functional form of these combinations when expressedin Cartesian coordinates as listed in the right-most column. Highly symmetric molecu-lar structures can be explained on the basis of the linear combination of orbitals. Forexample, the methane CH4 exhibits a spatial structure with the carbon atom at the centreand the four hydrogen atoms at the vertices of an equilateral tetrahedron. This structureis not explained by the p-orbitals of the central carbon atom alone but is due to a fur-ther orbital mixing. The l = 0 s-orbital combines with the three p-orbitals to producethe tetrahedral structure. This further mixing of s- and p-orbitals is called hybridisationbecause it involves linear combinations of solutions to the Schrödinger equation withdifferent angular momenta. As such, these ‘hybridised’ orbitals are not pure quantum an-gular momentum states, but they nevertheless appear in nature because the hybridisationlowers the energy of the molecule. Highly symmetric molecules like methane CH4 or sul-phur hexafluoride SF6 do not exhibit dipole absorption or emission between rotationalor vibrational states because they have no permanent dipole moment. However, a simplesubstitution of a chlorine atom for one of the hydrogen atoms in methane, for example,yields chloromethane CH3Cl, which does have a permanent dipole moment. Rotationand vibration of the molecular structure results in oscillation of the dipole, which in turnpermits rotation-vibration transitions through a dipolar interaction with incident light.We will examine in more detail this type of molecular spectra for simple diatomic mol-ecules when we discuss the rigid-rotor, harmonic oscillator model of molecular spectrain Section 12.9.

12.7 Interaction of light with the hydrogen atom

12.7.1 Semiclassical absorption and emission

We will first consider a two-level system is order to keep the notation simple and theideas clear. Once we have expressions for the rates of absorption and emission we will

Interaction of light with the hydrogen atom 347

apply the result to the transitions between the hydrogen atom ground state and the firstexcited state.

We start by returning to the time-dependent Schrödinger equation, Equation 12.18,rewritten here for convenience:

H�n(r, t) = ih∂�n(r, t)

∂t(12.98)

The Hamiltonian operator now consists of two terms: the first term is the Hamiltonianof the atom itself that we developed earlier in the chapter, Hatom. To this atomic Ham-iltonian, we add a time-dependent term, V (t), which will be closely related to a drivingelectromagnetic field, with frequency ω0. The total Hamiltonian is the sum of these twoterms. Now we write the stationary state n of the atom as

�n(r, t) = e–iEnt/hψn(r) = e–iωntψn(r) (12.99)

where we have used

En = hωn (12.100)

the Planck relation between quantised energy and frequency (see Chapter 6). For thecase of the hydrogen atom, the label n can be considered the principal quantum numbern discussed in Section 12.4.1. The time-independent Schrödinger equation becomes

Hatomψn(r) – Enψn(r) = 0 (12.101)

For the moment we consider the hydrogen atom as a two-level system and write

Hatomψ1 = E1ψ1 = hω1ψ1

Hatomψ2 = E2ψ1 = hω1ψ2

with

hω0 = h (ω2 – ω1)) = E2 – E1

We now add the V (t) term, the interaction energy of the atom with the driving field withfrequency close to ω0:

H = Hatom + V (t) (12.102)

With the field turned on the state of the system is no longer stationary and becomes acoupled, time-dependent linear combination of the two stationary states:

�(r, t) = C1(t)ψ1e–iω1t +C2(t)ψ2e–iω2t (12.103)


with C1,C2 the coupling coefficients. We require �(r, t) to be normalised:∫|�(r, t)|2 dV = |C1(t)|2 + |C2(t)|2 = 1

Now, if we substitute the time-dependent solution, Equation 12.103, back into thetime-dependent Schrödinger equation, Equation 12.98, multiply from the left with thecomplex conjugate, ψ∗1 e

iωt, and integrate over all space, we obtain

C1

∫ψ∗1 Vψ1 dr +C2e–iω0t

∫ψ∗1 Vψ2 dr = ih

dC1

dt(12.104)

To simplify notation we write the integral ‘matrix elements’∫ψ∗1 Vψ1 dr and

∫ψ∗1 Vψ2 dr

as V11 and V12. So we have,

C1V11 +C2e–iω0tV12 = ihdC1

dt(12.105)

and similarly for C2 we get

C1eiω0tV21 +C2V22 = ihdC2

dt(12.106)

These two equations define the time evolution of the atom-plus-field system as the atomundergoes a stimulated change between states 1 and 2. We need to solve this pair ofequations for C1(t) and C2(t). The squares of the absolute values of these coefficientscan be interpreted as the probability of finding the atom in either stationary state 1 or 2,as a function of time. Therefore, we are really more interested in |C1(t)|2 , |C2(t)|2 thanthe coefficients themselves.

12.7.1.1 Atom-field coupling operator

We now turn our attention to the details of V (t). This term is an interaction energy oper-ator, and from the classical theory of electrostatics we know that the leading interactionV between an electric field and neutral matter is the dipole term:

V = –p · E (12.107)

where p is the dipole moment and E is the electric field of, say, a propagatingelectromagnetic wave through a polarisable material. The classical dipole itself isdefined as

p =∑i

pi =∑i

qiri (12.108)

where qi is the charge at position ri with respect to the origin of the coordinate system(see Chapter 5). In the case of a classical atom with one electron orbiting a massiveproton, the instantaneous classical dipole can be taken as


p = –er (12.109)

where e is the charge on the electron and r is the radial position of the electron withrespect to the origin, centred at the proton. If an external E-field is now applied to theclassical atom, the interaction energy is given by

V = –p · E = e r · E (12.110)

Passing to the quantal atom, we use the correspondence principle to write the r and poperators:

r→ r (12.111)

p→ p (12.112)

but we do not quantise the E-field. The rate of atomic absorption and emission we willobtain is based on this semiclassical approximation in which the atom is quantised butthe field remains classical. We write the dipole interaction operator as

V = –p(r) · E (12.113)

= e r · E (12.114)

where r = r is the electron radial coordinate centred at the proton. The operator Vexhibits odd parity with respect to this electron coordinate so that the matrix elementsV11 and V22 in Equations 12.105 and 12.106 will vanish. Only atomic states of oppositeparity can be coupled by the dipole interaction. For the E-field we write the real form,

E = eE0 cos(ωt – kz)

were e is the polarisation unit vector, E0 the field amplitude, ω the frequency, and kthe propagation parameter. We have chosen the field to be propagating along z. Theinteraction takes place over the spatial extent of the atom, ∼ a0, which is only about0.5 nm. For electromagnetic fields ranging in wavelength from the near ultraviolet to thenear infrared, the wavelength is hundreds of nanometres. Therefore, the kz term in thefield argument is negligible and the E-field can be considered effectively constant overatomic dimensions. The absolute interaction matrix element can therefore be written as

|V12(t)| = |V12| cosωt=[∫

ψ∗1 (r)erψ2(r) dV]E0 cosωt

= p12E0 cosωt (12.115)


Since the dipole-field interaction is an energy, we can also write it as

|V12(t)| = h�0 cosωt

with the Rabi frequency given by

�0 =p12E0

h(12.116)

12.7.2 Selection rules for hydrogen atom transitions

In classical electromagnetic theory, an oscillating charged dipole can radiate a continuousspectrum. We found in Chapter 2, Equation 2.157 that the power emitted by a classicaldipole varies as the fourth power of the oscillation frequency ω, on which there is norestriction. In contrast, the absorption and emission of radiation by atoms only occurs atcertain well-defined frequencies, giving rise to the characteristic atomic line spectrum.The rules that govern these allowed transition frequencies are called selection rules. Thequantisation of the emission and absorption of radiation is one of the principal differ-ences between classical and quantal light–matter interaction. We can understand theorigin of the selection rules by studying the quantal dipole transition moment p12:

p12 =∫ψ∗nlml (r, θ ,ϕ)erψn′ l′m′l (r, θ ,ϕ)r

2 sin θ dθ dϕ dr (12.117)

where the primes on the quantum number labels indicate a final state different from theinitial state. This dipole matrix element can be simplified by factoring the wave func-tions into their r, θ ,ϕ constituents and resolving the vector dipole moment into its x, y, zcomponents:

p12 = e∫

rR∗nl(r)Rn′ l′r2drP∗l (cos θ)Pl′(cos θ)e

–imlϕeiml′ϕ sin θ dθ dϕ (12.118)

pz = r cos θ (12.119)

px = r sin θ cosϕ (12.120)

py = r sin θ sinϕ (12.121)

First, considering pz, we see that this component is independent of ϕ. Therefore, thedipole transition matrix element will have a factor

pz ∼∫ 2π

0ei(ml–ml′ )ϕ dϕ (12.122)

Since the integral over the period of any oscillatory exponential function with imaginaryargument is zero, the only way this factor can be non-zero is if the argument of theexponential itself is zero. Therefore, we have the first selection rule:


ml –ml′ = �ml = 0 (12.123)

Next, still considering pz, we see that we will have another factor in the transition momentintegral,

pz ∼∫ π

0P∗l (cos θ)Pl′(cos θ) cos θ sin θ dθ (12.124)

Now, as we show in Appendix G, the Legendre functions Pl(cos θ) have definite parity.Functions labelled with even l are unchanged (even) with respect to the parity operationand functions labelled with odd l exhibit odd parity (change sign). The parity operationis sign reversal of the function variable. In the case of the Legendre functions the variableis cos θ . The parity operation is

cos θ → cos(π – θ) = – cos θ (12.125)

and the parity property of the Legendre functions is that

Pl[cos(π – θ)]→ Pl(cos θ) l = 0, 2, 4, . . .

pl[cos(π – θ)]→ –Pl(cos θ) l = 1, 3, 5, . . .

The integrand of the integral in Equation 12.124 must be even with respect to θ overthe limits of integration. Since the factor cos θ sin θ exhibits odd parity, the product PlPl ′must also be odd so as to render the overall integrand even. Therefore, we see that�l = 0 is a ‘forbidden’ transition and �l = ±1 is allowed, or at least not forbidden, bythe integrand symmetry. Turning now to px, py, we will have factors

px ∼∫ 2π

0e–imlϕ cosϕeiml′ϕ dϕ (12.126)

py ∼∫ 2π

0e–imlϕ sinϕeiml′ϕ dϕ (12.127)

and since cosϕ and sinϕ are linear combinations of eiϕ and e–iϕ , it is clear that in orderto avoid a null integral, another selection rule for these two components of the transitionmoment is

�ml = ±1 (12.128)

More generally we have to consider the overall parity operation that transforms the dipolevector er to –er. In polar coordinates this parity inversion is accomplished by

r→ –r (12.129)

θ → π – θ (12.130)

ϕ→ ϕ ± π (12.131)


5s

4s

3s

2s

1s

5p

4p

3p

2p

5d

4d

3d

5f

4f

Figure 12.4 Grotrian diagram for nl energy levels of the hydrogen atom. Thearrows indicate some of the allowed transitions. Even with the restrictiveselection rule �l = ±1, the complicated cascading transition from high-lyingenergy states to lower-lying states is evident.

We see from Equation 12.118 that the radial part of the integrand has odd parity,and therefore, the product of the Legendre functions must show odd parity as well.Therefore, l must differ from l ′ by ±1 or we have the selection rule

�l = ±1 (12.132)

Summarising, the selection rules for allowed dipole transitions in the hydrogen atom are

�l = ±1 (12.133)

�ml = ±1, 0 (12.134)

The various allowed atomic transitions are often summarised in a schematic chart calleda Grotrian diagram. Figure 12.4 shows a Grotrian diagram for some of the transitions inthe hydrogen atom. The notation for the levels derives from atomic spectroscopy. Theprincipal quantum number n is indicated as n = 1, 2, 3, 4, . . . but the l quantum numberis assigned a letter s, p, d, f , . . .. The letters indicate progressions of spectroscopic linesthat have some common property. The quantum number l = 1 is labelled s for ‘sharp’lines. The l = 2 levels are labelled p for ‘principal’ lines, l = 2→ d for ‘diffuse’ lines.

12.7.2.1 Calculation of atomic absorption and emission rates

Now let us go back to the coupled equations, Equations 12.105 and 12.106, and re-write them in terms of the Rabi frequency and ω0, the frequency difference between theground and excited states:


�0 cosωt e–iω0tC2 = idC1

dt(12.135)

�∗0 cosωt eiω0tC1 = i

dC2

dt(12.136)

We take the initial conditions (at t = 0) to be unit probability of finding the two-statesystem in state 1 and null probability for finding it in the excited state 2. At later timesthe probability for finding the system in state 2 is P = |C2(t)|2. At early times the rateof increase of this probability is

dPdt

=|C2(t)|2

t(12.137)

so our task is to calculate C2(t) at times just later than t = 0. Applying the initialconditions, it is not difficult to show that

C2(t) =�∗02

[1 – ei(ω0+ω)t

ω0 + ω+

1 – ei(ω0–ω)t

ω0 – ω

](12.138)

If the frequency ω of the driving wave approaches the two-state difference frequency ω0

it is quite clear that the first term in Equation 12.138 will be negligible compared to thesecond. Near resonance we can therefore write to good approximation:

C2(t) � �∗02

[1 – ei(ω0–ω)t

ω0 – ω

](12.139)

We have, therefore,

|C2(t)|2 = |�0|2 sin2 [(ω0 – ω)(t/2)]

(ω0 – ω)2(12.140)

and when ω→ ω0, around resonance, we obtain a limiting expression:

|C2(t)|2 =14|�0|2 t2 (12.141)

Thus, with ω near ω0 and at early times under weak-field conditions, the probability ratefor finding the system in the excited states increases linearly with time:

dPdt

=14|�0|2 t (12.142)

12.7.3 Finite spectral width of absorption

The incident driving field always has some spectral width. The source might be, forexample, a broad band arc lamp or the output from a monochromator coupled to a


synchrotron. Therefore we need to write the field energy as an integral over the spectralenergy density of the excitation source in the neighbourhood of the transition frequency:

12ε0E2

0 =∫ ω0+

12�ω

ω0–12�ω

ρω dω (12.143)

where ρω is the spectral energy density of the excitation source with a line widthof �ω. Note that the MKS units of spectral energy density are Jm–3s. Now fromEquation 12.140 we can write

|C2(t)|2 =(p12E0

h

)2 sin2 [(ω0 – ω)t/2]

(ω0 – ω)2(12.144)

and substituting from Equation 12.143 we have

|C2(t)|2 =2p212ε0h2

∫ ω0+12�ω

ω0–12�ω

ρω dω (12.145)

For conventional broad-band excitation sources we can safely assume that the spectraldensity is constant over the line width of the atomic transition and replace ρ(ω) in theintegrand with ρ(ω0) outside it. Assuming a continuous wave (cw) excitation source, theintegral can be evaluated:

∫ ω0+12�ω

ω0–12�ω

sin2 [(ω0 – ω)t/2]

(ω0 – ω)2dω =

π t2

(12.146)

So finally, the expression for the time rate of increase of the probability of finding thesystem in state 2 (i.e. the radiation absorption rate), integrated over the spectral width ofthe source, is

dPdt

=|C2(t)|2

t=πp212ε0h2

ρ(ω0) (12.147)

12.7.4 Comparison to Einstein B coefficient

Now, from Chapter 6 we know that the Einstein A and B coefficients represent spon-taneous emission and stimulated absorption under conditions of equilibrium betweena radiation field and a collection of material absorber-emitters. These coefficients wereintroduced into phenomenological rate equations and originally had no underlying inter-pretation of just how the light was absorbed or emitted. However, by recognising that theEinstein absorption rate, B12ρω, is equivalent to the semiclassical dP/dt we can writethe B coefficient in terms of the quantum mechanical dipole matrix element p12. Moresuccinctly:


dPdt

=πp212ε0h2

ρ(ω0) = B12ρ(ω0) (12.148)

or

B12 =πp212ε0h2

(12.149)

The p12 that appears in Equation 12.148 has a specific orientation with respect to thepolarisation of the exciting field. Equation 12.115 shows the angle θ of p12 with respectto the E-field of the exciting light, which is zero. However the Einstein B coefficientassumes isotropic radiation, so in order to really make the comparison of B and p12meaningful we have to average the right-hand side of Equation 12.149 over all angles.The average value of the square of the dipole moment matrix element is given by

⟨p212⟩= p212

⟨cos2 θ

⟩=

13p212 (12.150)

So, finally we have

B12 =πp2123ε0h2

(12.151)

We have restricted this development to the consideration of an artificial atom, thetwo-level system without any substructure or energy-level degeneracy. No real atomcorresponds to this simple scheme. Even the simplest atom, hydrogen, with one electronand one proton, exhibits orbital angular momentum degeneracy in the excited p level.However, it is a simple matter to take degeneracy into account in writing the expressionsfor B12,B21, and A21, the stimulated absorption, stimulated emission, and spontaneousemission coefficients, respectively. The stimulated absorption and emission coefficientsare related to the degeneracies g1, g2 of states 1, 2, respectively, by

g1B12 = g2B21 (12.152)

The spontaneous emission rate A21 is independent of the upper state degeneracy but isrelated to B21 by

A21 = B21hω3

0

π2c3(12.153)

Therefore, we have

B21 =g1g2B12 =

g1g2

π2c3

hω30

(12.154)


and, using Equation 12.151,

A21 =g1g2

ω30p

212

3πε0hc3(12.155)

Thus, if we can calculate or determine experimentally the transition dipole moment of aquantal atom we can obtain the spontaneous emission rate, and the stimulated emissionand absorption rate coefficients.

12.7.5 Interpretation of dipole radiation in quantal atoms

We saw in Chapter 2, Section 2.8, Equation 2.138 that the leading source of classicalelectromagnetic radiation is the harmonically oscillating charge dipole,

p(t) = p0 cos(ωt)z = q0a cos(ωt)z

where q0 is a point charge oscillating over a length a aligned along z with frequencyω. In classical electrodynamics an oscillating dipole is a tangible, physical entity withobservable properties. In quantum mechanics, the dipole, like all observables, is castas an abstract mathematical operator, p(t) = qr(t). How then does one connect theclassical picture with the quantal formulation? The answer is through the dipole ‘matrixelement’ or expression for the ‘average value’ of the dipole operator. The conventionalform for the matrix element is to sandwich the operator between the wave functionrepresenting the state of the dipole (on the right) and its complex conjugate (on theleft). The ‘sandwich’ is then integrated over all space:

⟨p⟩=∫�∗(r, t)p�(r, t) dV (12.156)

The rationale for this expression comes from the interpretation of the product of a wavefunction and its complex conjugate as a probability density P:

P = �∗(r, t)�(r, t)

Assuming the wave function is normalised,∫P dV =

∫�∗(r, t)�(r, t) dV = 1

In classical statistics the average value of any quantity is calculated by integrating theprobability distribution of values over the space of all possible values. A familiar exampleis the average speed v of a gas molecule subject to the Maxwell–Boltzmann distributionat fixed temperature T :

〈v〉 =∫

P(v)v dv =(

m2πkmbT

)3/2 ∫4πv2e–

12mv

2/(kmbT) · v dv


where kmb is the Maxwell–Boltzmann constant. Similarly, in quantum mechanics

⟨p(t)

⟩=∫�∗(r, t)�(r, t)p(t) dV (12.157)

but because p is an operator, it is conventionally placed to the left of � so that it can‘operate’ on it, left to right. Now, the state of the atom, when subject to a harmonicdriving field, is not one of the stationary states but a time-dependent linear combinationof them. In the two-level atom we have

�(r, t) = C1(t)ψ1(r)e–iω1t +C2(t)ψ2(r)e–iω2t (12.158)

with mixing coefficients C1,C2 subject to the normalisation condition

|C1|2 + |C2|2 = 1 (12.159)

The differential probability of finding the dipole p(r) = –er between r an r + dr is then

P p dV = �∗�p dV = –e�∗�r dV (12.160)

and the average value of the dipole moment is the integral of this differential probabilityover all space:

〈p(t)〉 = –e 〈r(t)〉 =∫ [

C∗1ψ∗1 (r)e

iω1t +C∗2ψ2eiω2t](–er)·[

C1ψ1(r)e–iω1t +C2ψ2(r)e–iω2t]dV

= 2 cos(ω0t)∫C1C∗2ψ

∗1 (r)(–er)ψ2(r) dV

= 2C12p0 cos(ω0t) (12.161)

with ω0 = ω2 – ω1 and C12 = C∗1C2 = C∗2C1 and

p0 =∫ψ∗1 (–er)ψ2 dV (12.162)

We see that the dipole matrix element oscillates in time at frequency ω0 = ω2 – ω1 andwith an amplitude controlled by the coefficients C1,C2. When the two stationary statesare strongly coupled to form the time-dependent oscillating states, the dipole transitionis strong.


12.8 The fourth quantum number: intrinsic spin

Up to this point we have considered only the quantum numbers n, l,ml . We have usedthese indices to identify properties of the stationary states of the hydrogen atom. Thenumber n indexes the energy of the state, the number l the magnitude of its orbitalangular momentum, and ml the projection of the angular momentum on the axis ofquantisation; specifying, within the limits of the uncertainty principle, the direction inspace to which the orbital angular momentum orients. In addition to these three, a fourthquantum number ms exists. This quantum labels the orientation of the intrinsic spinangular momentum S of the electron. This spin angular momentum has no classicalanalogue. It does not correspond to the classical magnetic moment of a spinning elec-tron and it does not correspond to the magnetic field of an electron circulating in a Bohrorbit. It arises naturally from the theory of quantum electrodynamics (QED), which is,unfortunately, beyond the scope of this book. Every electron carries a quantity of in-trinsic spin S = h

√s(s + 1) where the spin quantum number s = 1/2. Therefore, the

magnitude of spin angular momentum intrinsic to every electron is S = (√3/2)h. Analo-

gous to the orbital angular momentum, the spin direction is given by the projectionof S on the quantisation axis. For the electron spin there are only two possible values,ms = ±1/2, and the spin state is labelled by this quantum number. A complete speci-fication of a stationary state of the hydrogen atom is therefore given by four quantumnumbers: n, l,ml ,ms. The quantum charge-dipole operator does not couple states of dif-ferent intrinsic spin, and so the quantum number ms remains unchanged during electricdipole transitions in the hydrogen atom.1 Our last selection rule is therefore �ms = 0.

In fact, stationary states of the hydrogen atom with l �= 0 do exhibit a magneticmoment that can be interpreted as arising from the electron orbital motion. The orbitingelectron is a current, and Ampère’s law tells us that a magnetic field is always producedat right angles to this loop current. This orbital magnetic is not the intrinsic spin, but inmulti-electron atoms the two magnetic dipoles, orbital and spin, can couple. The resultsof this coupling can be observed in the atomic transition spectra, but these considerationsare the proper domain of atomic structure and spectroscopy and will not be consideredfurther here.

12.9 Other simple quantum dipolar systems

12.9.1 Rigid rotor model of a diatomic molecule

A heteronuclear diatomic molecule has an asymmetric charge density localised near thetwo nuclei comprising the molecule. This charge asymmetry produces a net charge sep-arated by internuclear distance of the molecule, and therefore a permanent dipole. In the

1 In multi-electron atoms the orbital and spin angular momenta may couple strongly. When this ‘spin-orbitcoupling’ becomes significant, the individual angular momenta L and S are no longer conserved, and only thetotal angular momentum J = L+S specifies the state of the system. Further discussion of angular momentumcoupling in multi-electron systems is, however, beyond the scope of what we can discuss here.

Other simple quantum dipolar systems 359

rigid rotator model, we ignore the molecular vibration and consider only the molecularrotation in space. The projection of a rotating dipole on any axis contained in the planeof rotation reveals an effective dipole oscillation, and therefore, the possibility of absorp-tion and emission of dipole radiation. Note that this dipolar oscillation is due to nuclearmotion, not the electron orbital motion. The nuclei of molecules move much more slowlythan the electrons in the charge density binding the nuclei together. Because of this greatdifference between effective nuclear and electron velocities, their motions are, in manycases, effectively decoupled. The energy of the molecule can then be calculated as thesum of the rotational, vibrational, and electronic energy; and we can write down a Schrö-dinger equation for each type of motion, solve them separately, and write the total wavefunction as a product of three. Thus:

Emolecule � Erot + Evib + Eelec (12.163)

ψmolecule � ψrotψvibψelec (12.164)

This motional decoupling is called the Born-Oppenheimer approximation.Figure 12.5 shows the setup for modelling the rotational motion. The classical rota-

tional energy is usually expressed in terms of the angular frequency ω and the momentof inertia, I :

Erot =12Iω2 (12.165)

The moment of inertia is given by

I = m1r21 +m2r22 (12.166)

where

r1 =m1

m1 +m2r and r2 =

m2

m1 +m2r (12.167)

m2

Rigid rotor

r2

r

r1

m1

Figure 12.5 Rigid rotor model of a het-eronuclear diatomic molecule. Internuclearseparation r is considered constant. Massm2>m1. Permanent electric dipole isaligned along r and rotates with themolecule.


Now substitute Equation 12.167 into Equation 12.166. The result is

I =m1m2

m1 +m2r2 (12.168)

and identify the reduced mass μ as

μ =m1m2

m1 +m2(12.169)

We now have reduced the problem to one mass and one coordinate,

I = μr2 (12.170)

and the energy of rotation can be written in terms of the rigid rotor angular momen-tum, J:

J = Iω (12.171)

and

Erot =J2

2I(12.172)

The quantum rigid rotor problem is setup using the correspondence principle be-tween classical dynamical variables and their quantum operator counterparts. FromEquation 12.172 we draw the following correspondences,

Erot → Hrot (12.173)

J→ J = –h2∇2 (12.174)

r→ r = r (12.175)

Then we write the time-independent Schrödinger equation,

[Hrot – Erot

]ψrot = 0 (12.176)[

∇2 +2I

h2

]ψrot = 0 (12.177)

The solutions to Equation 12.177 are analogous to the orbital angular momentum of theelectron in the hydrogen atom problem. There is a family of solutions:

ψrot = PJ(cos θ)eimJϕ = Y

mjJ (θ ,ϕ) (12.178)


Rigid Rotor Energy Levels

Erot

5

4

3

2

10J Figure 12.6 Energy level diagram of the rigid rotor eigenstates.

where PJ(cos θ) are again the associated Legendre functions and YmjJ are the spherical

harmonics. The eigen energies are

Erot =h2J(J + 1)

2I; J = 0, 1, 2, . . . (12.179)

where J are the rigid-rotor angular momentum quantum numbers (analogous to l for thehydrogen atom). The selection rules on rigid rotor transitions are similar to the selectionrules in hydrogen:

�mJ = 0,±1; �J = ±1 (12.180)

Figure 12.6 shows a diagram of the energy levels and pure rotational transitions in therigid rotor model.

In the realm of molecular spectroscopy the energy levels are often classified accordingto terms, by which is meant that the transition energy is expressed in units of reciprocallength. Thus, Equation 12.179 is expressed as

F(J) =Erot(J)hc

=h

8π2cIJ(J + 1) = BJ(J + 1) (12.181)

where h, c have their usual meanings and B is called the rotational constant. It charac-terises the rotational energy in units of reciprocal length. Traditionally in spectroscopythe conventional unit is cm–1, called a ‘wavenumber’. Pure rotational transitions typicallyhave rotational constants of some tens of cm–1. For example, the rotational constant ofhydrogen iodide, HI, is about 6.5 cm–1 and that of H2 is about 61 cm–1.

12.9.2 Harmonic oscillator model of a diatomic molecule

For heteronuclear diatomic molecules, a permanent dipole moment exists, orientedalong the internuclear axis. Harmonic molecular vibrations along this axis will result


m2

r2

rermin

rmax

r1

Harmonic Oscillator

m1

Figure 12.7 Harmonic oscillator model of molecular vi-bration along the internuclear axis r. If m1 �= m2 then acharge dipole exists along the axis, the amplitude of whichwill vary with the oscillation of r.

in a dipole oscillation at the same frequency, and therefore quantised dipole emissionand absorption can take place. Figure 12.7 shows the harmonic oscillator model of theheteronuclear diatomic molecule. The energy of a harmonic oscillator is the sum of itskinetic and potential energies. From elementary treatments we know that the kinetic en-ergy is p2/2μ, where p is the linear momentum of the reduced mass μ, and the potentialenergy V (r) is given by

V (r) =12kr2 (12.182)

where k is the force constant given by

k = μω2 (12.183)

with ω the angular frequency of oscillation. Once again we use the correspondenceprinciple to form the Hamiltonian operator:

p→ p = –ih∂

∂r(12.184)

r→ r = r (12.185)

Evib→ Hvib =p2

2μ+

12kr2 (12.186)

The harmonic oscillator Schrödinger equation then becomes[Hvib – Evib

]ψvib = 0 (12.187)

The eigen energies take on the form

Evib = hω(v +

12

); v = 0, 1, 2, . . . (12.188)

and the eigenfunctions are closely related to the Hermite polynomialsHv(√αr) discussed

in Appendix H. The physically admissible solutions to Equation 12.187 are


Harmonic Oscillator Energy Levels

V(r)

01

234

5

r

Figure 12.8 Energy level diagram showing allowed energies of thequantum harmonic oscillator. Note that the ground state energyE0 = 1/2hω.

ψvib = NnormHv(√αr)e–μω/hx

2(12.189)

where Nnorm = 1/√2vπ1/2v! is a normalisation factor and α = μω/h is a scaling factor

for the coordinate r. Figure 12.8 shows the equally spaced ladder of energies Evib = hωstarting from the ground state with zero-point energy 1/2hω.

In spectroscopy, the energy of vibrational transitions are expressed as vibrational termsanalogous to the spectroscopy of pure rotations. The vibrational terms are expressed as

G(v) =Evib

hc= ω

(v +

12

); v = 0, 1, 2, . . . (12.190)

where ω means the vibrational energy, usually reported in units of cm–1. This vibra-tional ω should not be confused with the angular frequency of some wave or transition.Unfortunately, spectroscopic notation became entrenched long before the modern as-sociation of ω with a unit of angular frequency. The vibrational transitions of diatomicmolecules exhibit much greater energy than rotational transitions. For example, the vi-brational termG(v) in HI is 2309 cm–1, a factor of about 350 greater than the rotationalterm.

12.9.3 Electronic states of molecules

In addition the rotational and vibrational transitions, arising from the motion of per-manent dipoles, molecules can undergo electronic transitions, somewhat analogous tothe transitions we have studied in the hydrogen atom. Diatomic molecules have lowersymmetry (cylindrical) than atoms (spherical), and therefore the great simplification ofvariable separation is not so easy to carry out. Nevertheless, one can characterise mo-lecular orbitals and assign symmetry quantum number to electronic states. The properstudy of molecular structure and spectroscopy is outside the scope of this book, but suf-fice it to state that molecular electronic transitions do occur and their energies are againmuch greater than pure vibrational or rotational transitions. Continuing with the ex-ample of HI, the first electronic transition between molecular bound states is the X → Btransition. Here, X indicates a molecular ground electronic state and B labels a bound


Table 12.5 Molecular transitions, term energies, and frequencies.

Transition Type Term Energy (cm–1) Frequency Range

Rotation 3–30 EHF RadioVibration 1000–3000 THz-IRElectronic 10 000–30 000 Vis-UV

excited electronic state. The electronic spectroscopic term for this transition is about5.6 × 103 cm–1, about 25 times greater than the vibrational term. Hierarchy of Tran-sitions Rotation, vibration, and electronic transitions follow an energy hierarchy fromrelatively low to high. Table 12.5 summarises the typical energies for these dipole tran-sitions. Rotational spectroscopy can be studied with microwave technology, but purevibrational transitions (at least for ‘typical’ diatomic molecules) fall in an awkward partof the electromagnetic spectrum. Radiation sources and detectors in the THz (1012 Hzand far infrared (IR) regimes are not, at present, a well-developed and readily avail-able technology. Electronic spectroscopy, in contrast, can be studied with a vast array ofsources, optical dispersion instrumentation, and detectors. Of course, when a moleculeabsorbs or emits light in the visible-ultraviolet range it undergoes a ‘rovibronic’ transi-tion in which all three types of internal molecular motion change. Figure 12.9 shows,schematically, how these rovibronic transitions couple lower states to higher states. Therotation-vibration motion gives rise to underlying manifolds of transition lines, riding ontop of the electronic transition. The amplitude and spacing of the transition lines withinthe manifold provides information on vibrational and rotational motion.

ElectronicExcited State

ElectronicGround State

Figure 12.9 Schematic of rovibronic transitions where radiationconnects a ground-state manifold of vibration-rotation states to anexcited-state manifold. Separations between states are not to scale.

Exercises 365

12.10 Summary

This chapter again starts with the most elementary models of harmonic oscillation andradiative damping as a warm-up to the main event, the Schrödinger equation for thehydrogen atom. The full 3-D Hamiltonian is presented, the equation solved in polarcoordinates, and the solutions for radial space coordinate r, and polar (θ), azimuthal(ϕ) angular coordinates established. The final 3-D wave function is then the product ofthe three solutions. The nature of the bound states leading to the electron continuumis discussed together with the significance of the n, l,ml quantum numbers. Then realorbitals, constructed from linear combinations of the complex product solutions (hy-bridisation), are developed and shown to be useful in theories of molecular structure andchemical binding. Dipole radiative emission and absorption is the next topic where thesymmetries of the radial and angular solutions, together with the symmetry of the dipolecoupling, are used to determine transition quantum number ‘selection rules’. The chap-ter ends with a brief discussion of the ‘fourth quantum number’, magnetic dipole spin,and the structures and spectroscopy (rotational, vibrational and electronic) of diatomicmolecules.

12.11 Exercises

1. Calculate the average value of the electron radius for the hydrogen atom in its groundstate, R10. Use the expression from Table 12.1.

2. For the same state, calculate the maximum probability for finding the electron atsome distance from the proton.

3. Explain why the average value for the electron radius and the most probable valuefor the electron radius are not equal.

4. Given the ground state and first excited states of the hydrogen as

ψ100 =1√π

1

a3/20

e–r/a0

and

ψ210 =1√π

1(2a0)5/2

re–r/2a0 cos θ

show that

p12 =215/2ea0

35

where e is the charge on the electron and a0 is the Bohr radius.


5. The energy-level spacing for this first transition in hydrogen is called the Lyman αtransition. The wavelength at line centre is λ = 121.7 nm. Calculate the spontaneousemission rate A21 (s–1) for this transition.


1. J.-P. Pérez, R. Carles, and R. Fleckinger, Électromagnétism, Fondements et applications,3ème édition, Masson (1997).

2. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley (1977).

3. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th edition,Elsevier Academic Press (2005).

4. R. Loudon, The Quantum Theory of Light, 3rd edition, Oxford University Press(2003).

5. A. Corney, Atomic and Laser Spectroscopy, Clarendon Press, Oxford (1977).

6. J. Weiner and P.-T. Ho, Light-Matter Interaction: Fundamentals and Applications,Wiley-Interscience (2003).

7. G. Herzberg, Molecular Spectra and Molecular Structure; I Spectra of DiatomicMolecules, Van Nostrand Reinhold (1950).

Appendix ANumerical Constants and Dimensions

A.1 Numerical values of some fundamental constants

Permittivity of free space ε0

ε0 = 8.854× 10–12 farad m–1 (A.1)

Permeability of free space μ0

μ0 = 4π × 10–7 = 1.257× 10–6 henry m–1 (A.2)

Speed of light

c =1√μ0ε0

= 2.998× 108 metres s–1 (A.3)

Resistance of free space Z

Z =√μ0

ε0= 376.7 ohms (A.4)

Boltzmann constant kB

kB = 1.38065× 10–23 J K–1 (A.5)

Stefan–Boltzmann constant σ

σ = 5.670× 10–8 W m–2 K–4 (A.6)

Atomic mass unit amu

amu = 1.660× 10–27 kg (A.7)

Planck constant h

h = 1.054572× 10–34 J s (A.8)

Elementary charge e

e = 1.602176× 10–19 C (A.9)

368 Appendix A: Numerical Constants and Dimensions

Bohr radius a0

a0 = 0.529177×–10 m (A.10)

Bohr magneton μB

μB = 927.401× 10–26 J T–1 (A.11)

Electron volt eV

eV = 1.602176× 10–19 J (A.12)

A.2 Dimensions of electromagnetic quantities

Fundamental quantities are chosen to be massM, length L, time T , and charge Q. Theunits are the SI units of kg, m, s, and C, respectively.

Quantity Symbol Dimensions SI unit

Force F MLT–2 NewtonEnergy E,E ML2T–2 JoulePower W ML2T–3 WattField energy flux S MT–3 Watt per square metreField momentum flux g ML–2T–1 Joule per cubic metre ·

metre per secondCharge q Q CoulombCurrent I QT–1 AmpèreCharge density ρ QL–3 Coulomb per cubic metreCurrent density J QT–1M–2 Ampère per square metreResistance R ML2T–1Q–2 OhmConductivity σ M–1L–3TQ2 (Ohm·metre)–1

Electric potential V ML2T–2Q–1 VoltElectric field E MLT–2Q–1 Volt per metreCapacitance C M–1L–2T2Q2 FaradDisplacement field D QL–2 Coulomb per square metrePermittivity ε M–1L–3T2Q2 Farad per metreElectric dipole moment p LQ Coulomb·metreMagnetic flux ML2T–1Q–1 WeberMagnetic induction field B MT–1Q–1 Weber per square metreMagnetic field H QT–1L–1 Ampère per metreInductance L ML2Q–2 HenryPermeability μ MLQ–2 Henry per metreMagnetic dipole moment m L2T–1Q Ampère·square metre

Appendix BSystems of Units in Electromagnetism

B.1 General discussion of units and dimensions

We conventionally decide to take the three mechanical units of mass m, length l, andtime t, as three fundamental, basic units. However, this choice is just for convenience.The number of units need not be three and they need not be these three. For example,we could increase the number of units by defining the kinetic energy as

E = k ·m ·(lt

)2

= kv2 (B.1)

where k is a constant of proportionality with some dimensions, say, the ratio of chargeto mass of an electron, (e/me). We could define

k = 2eme

(B.2)

Then kinetic energy would be expressed by

E =(eme

)v2 (B.3)

and would have units of l2/qt2, where q is some unit of electrical charge. The number ofbasic units is now four: mass, length, time, and electric charge. We could also decreasethe number of units by setting some constants equal to unity and without dimensions.For example, setting me = e = h = 1 is such a choice, sometimes called atomic units, andconvenient in atomic and molecular quantum mechanical expressions.

Here is another example: the volume can be defined as function of the length unit,

V = kl3 (B.4)

where k is an arbitrary constant, usually set equal to unity. But suppose we had someproblem dealing only with spherical volumes,

V = k43πr3 (B.5)

370 Appendix B: Systems of Units in Electromagnetism

We could define k to be 34π which would make the volume read

V = r3 (B.6)

in some system of ‘rationalised’ units.Now electromagnetic (E-M) theory differs from classical mechanics in that a funda-

mental constant, ‘the speed of light’, c appears explicitly. This constant has dimensions,l/t, and if these units are changed, the appearance of E-M expression will change as well.We will see how the speed of light makes its appearance by comparing electrostatic andmagnetostatic forces.

B.2 Coulomb’s law

Suppose we start with Coulomb’s law, where ‘force’ has been defined in terms ofmass, length, and time but the constant of proportionality, K1, or the units of electricalcharge, q, have not yet been fixed:

F = K1 ·(q1q2r2

)(B.7)

We now choose to set K1 = 1, which then determines the dimensionality of the chargeto be l3/2m1/2/t. This choice of proportionality constant in Coulomb’s law is used inthe esu (electrostatic units) system of units. The usual units of mass, length, and timein this system are the gram, centimetre, and second. The unit of charge is called the‘statcoulomb’.

We can use Coulomb’s law to write the electric field as ‘force per unit charge’,

Fq1

= E =q2r2

(B.8)

Or more properly in vector form,

E = qrr3→[∫

Vρ(x)d3x

]rr3

(B.9)

where r is the unit vector pointing in the E direction, and ρ(x) is the charge density.Now from Gauss’s law we know that∮

SE · n da = 4π

∫Vρ(x)d3x (B.10)

and from the divergence theorem,∮SE · n da =

∫V

∇ · E d3x (B.11)

from which we get the differential form of Gauss’s law,

∇ · E = 4πρ (B.12)

Appendix B: Systems of Units in Electromagnetism 371

But instead of choosing K1 = 1 in Equation B.7 we could have chosen K1 = 14π , which

would make the 4π factor in Equations B.10 and B.12 disappear. Such a choice of scaledunits might be called ‘rationalised’ units.

Up to now we have only considered E-M fields in a vacuum, but when we generaliseto dielectric media, the electric vector field E becomes the displacement vector D and isexpressed as

D = ε0E + κP (B.13)

The vector field P is called the ‘polarisation’ and represents the density of dipole mo-ments in the material. The constants ε0 and κ are proportionality constants. Theseconstants are not the same in esu and MKS units. In the esu system the displacementvector field is

D = E + 4πP (B.14)

In this system ε0 is chosen to be a pure number with unit value. This choice implies thatD, E, and P all have the same dimensions.

In the MKS system

D = ε0E + P (B.15)

As usual, the 4π factor is eliminated, but the factor ε0 appears explicitly. Note fromEquation B.7 and the choice of K1 = 1/ (4πε0) that this factor in the MKS system hasdimensions of q2t2/ml3 and units of ‘Coulomb squared per Joule metre’ (C2/J ·m), so theelectric field vector E does not have the same dimensions as the displacement vector Dand the polarisation vector P. To be consistent with the displacement field in polarisablematerials, the displacement field in free space is just written as

D0 = ε0E0 (B.16)

and P is zero since we do not consider free space to be polarisable (at least in classicalelectrodynamics). Now, going back to the differential form of Gauss’s law (Equa-tion B.12) and inserting ε0 on both sides, we have the generalisation to the displacementfield form that gives us

∇ ·D0 = 4πε0ρ (B.17)

But again, instead of choosing K1 = 1 as we did in Equation B.7, we could have chosenK1 = 1

4πε0, which would result in the disappearance of messy constants in the final

expression for Gauss’s law:

∇ ·D0 = ρ and ∇ · E0 =1ε0ρ (B.18)

This choice for K1 forms the basis for the ‘rationalised MKS’ system of units.


B.3 Ampère’s law

Currents I1, I2 moving in two parallel wires separated by some known distance dalso give rise to a force. The force equation, analogous to the Coulomb force inelectrostatics, is

dFdl

= 2K2I1I2d

(B.19)

where K2 is a proportionality constant whose value depends on the choice of units. Notethat here the force is ‘per unit length’ of the infinitely long wires. Now, from Equa-tions B.7 and B.19 we can do a dimensional analysis of the ratio, K1/K2. The result isthat the ratio of the two constants has units of velocity squared (l/t)2. It turns out frommeasurement that for equal electrostatic and magnetostatic forces the value of this ratio isequivalent to the speed of light in vacuum squared, c2. So we have K1/K2 = c2.

The magnetic induction field B is defined as the ‘force per unit current’ analogous tothe ‘force per unit charge’ that defines the electric field E:

B = 2K2Id

(B.20)

However, Equation B.20 is not quite the whole story. A defining relation between E andB would imply that B is just proportional to the ‘force per unit current’ so that, strictlyspeaking, we should write

B = 2K2K3Id

(B.21)

and then make some convenience argument about the choice of K3. In order to reallyset K3 correctly we need the equation that relates E to B. Such an equation exists. It isFaraday’s law of induction:

∇ · E = –K3∂B∂t

(B.22)

Table B.1 Common systems of units in electromagnetism.

Name Units K1 K2 Remarks

Electrostatic cgs 1 1/c2 charge unit statcoulomb

Electromagnetic cgs c2 1 current unit abampere

Heaviside–Lorentz cgs 1/4π 1/(4πc2) rationalized emu

Gaussian cgs 1 1/c2 convenient formally

SI m,kg,s 1/(4πε0) μ0/4π convenient numerically

Appendix B: Systems of Units in Electromagnetism 373

Notice that the Faraday relates a spatial derivative of E to a time derivative of B.Suppose, just to keep things simple, that E and B are the field components of a planewave. Then we have immediately the scalar relation,

ikE = K3ωB or E = K3

(ωk

)B (B.23)

The factor ω/k has units of l/t or velocity. We have, therefore, two easy choices for K3.If we set K3 = 1, then B = (1/v)E, and in free space B = (1/c)E. We could also setK3 = 1/c, in which case B = E.

We see that in the esu system of units the choice of K1=1 means that K2 = c2. In therationalised MKS system K1 = 1/(4πε0). The constant K2 has to be chosen so that theratio is c2. The choice is the following:

K1 =1

4πε0= 10–7c2 and K2 =

μ0

4π= 10–7 (B.24)

where the ‘permeability of free space’ μ0 has been introduced. The permeability servesthe same function for magnetic fields that permittivity serves for electric fields. The dif-ference between the Electrostatic system and the Gaussian system is in the choice of K3.In the Electrostatic system of units K3 = 1 while in the Gaussian system K3 = 1/c. The

Table B.2 Electromagnetic symbols and units

Quantity Symbol Units

Electric charge q C

Electric charge density ρ Cm–3

Electric current density J Am–2

Electric current I A

Electric conductivity σ Sm–1

Electric permittivity ε Fm–1

Electric susceptibility χe unitless

Electric field E Vm–1

Displacement field D Cm–2

Magnetic permeability μ Hm–1

Magnetic susceptibility χm unitless

Magnetic field H Am–1

Magnetic induction field B Vsm–2


Heaviside–Lorentz system is similar to the Gaussian except it is ‘rationalised’ therebyeliminating factors of 4π that often appear in the field equations of the Gaussian,Electrostatic, and Electromagnetic systems.

The magnetic fields in matter, analogous to Equations B.14 and B.15, are

H = c2B – 4πM (esu) and H =1μ0

B –M (MKS) (B.25)

The vector field H is, strictly speaking, the magnetic field, while B should always bereferred to as themagnetic induction field. Obviously, the magnetic fieldH plays an analo-gous role to the displacement field D in the macroscopic equations of electrostatics. Thevector field M is called the ‘magnetisation’ and plays a role analogous to P, the polarisa-tion. Note that neither in the esu system nor in the MKS system does B have the sameunits as H.

Note finally that the choice of K2 = μ0/4π in the MKS system implies that

1ε0μ0

= c2 (B.26)

Many of the quantities, symbols, and units (in the rationalised MKS system) arecompiled in Table B.2.

Appendix CReview of Vector Calculus

C.1 Vectors

A spatial vector is essentially a geometric entity comprising a magnitude (or length)and a direction. A scalar is some entity characterised only by magnitude. Many phys-ical quantities can be represented by scalars and vectors. As an example, FigureC.1shows an object of mass mmoving in a curved trajectory with velocity v and subject to aforce F. The acceleration ac points toward the origin of the central force. The distancefrom the object to the force centre is R. In this example m and R are scalars with onlyone number (amplitude) associated with these physical quantities. The vector quantities,v, F, ac have two associated properties: the amplitude and the direction. The geometricentity of the vector can be represented by the coordinates of the two end points. If thecoordinate system is transformed to some other one, the vector remains unchanged butits representation changes with the coordinate values of the endpoints. For example, thedirection can be represented by two numbers along a line in one-dimensional (1-D)space, two pairs of numbers in two-dimensional (2-D) space, and a pair of three num-bers in a three-dimensional (3-D) space. Very often the ‘tail’ of the vector is referred tothe origin of the coordinate system. The 3-D velocity vector, for example, in Cartesiancoordinates takes on the form

v = vxx + vyy + vzz (C.1)

where the unit vectors in the x, y, z directions are x, y, z and the magnitude of the vectorcomponents in the x, y, z directions are vx, vy, vz. The overall length or amplitude of the

vector is |v| =√v2x + v2y + v2z. A vector can also be expressed by a column of numbers,

each element corresponding to the component along a Cartesian coordinate axis:

v =

⎡⎣ vxvyvz

⎤⎦ (C.2)

A vector can be rotated within a coordinate space, or alternatively, the coordinate axesthemselves can be rotated (transformed so that the vector is represented in a new co-ordinate space). The rotation operation can be represented by a matrix. For a 2-Danticlockwise rotation we have, for example,

376 Appendix C: Review of Vector Calculus

m

Rvac F

Figure C.1 Examples of vector quantities andscalar quantities. Velocity v, force F, and accel-eration ac are vectors. The distance to the originR and the mass m are scalars.

v′ =[v′xv′y

]=[cos θ – sin θsin θ cos θ

]·[vxvy

](C.3)

Writing out the matrix multiplication explicitly we have

v′x = cos θ vx – sin θ vy (C.4)

v′y = sin θ vx + cos θ vy (C.5)

and FigureC.2 shows the coordinate rotation. This type of coordinate transformation ofa vector is called a unitary transformation because the length of the vector is preserved.This property is easy to see by taking the absolute value of the original vector and thetransformed vector. The length of the original vector is given by

|v| = (v2x + v2y)1/2 (C.6)

and the length of the rotated vector is

|v′| =(v′x

2 + v′y2)1/2

=[(cos2 θ + sin2 θ

) (v2x + v

2y

)]1/2=(v2x + v

2y

)1/2= |v| (C.7)

Vx cos θ

Vy cos θ θ

θ

VX

y

Vy

Vy´

VX´

V

X

X´

y´

Figure C.2 Coordinate rotation of coord-inate axes x – y to coordinate axes x′ – y′ byangle θ anticlockwise around vector V. Pro-jections onto x–y and x′ –y′ coordinate axesare shown.

Appendix C: Review of Vector Calculus 377

In 3-D we can represent the rotation matrix about the three Cartesian coordinate axes as

Rx(θ) =

⎡⎣1 0 00 cos θ – sin θ0 sin θ cos θ

⎤⎦ (C.8)

Ry(θ) =

⎡⎣ cos θ 0 sin θ

0 1 0– sin θ 0 cos θ

⎤⎦ (C.9)

Rz(θ) =

⎡⎣ cos θ – sin θ 0sin θ cos θ 00 0 1

⎤⎦ (C.10)

These matrices rotate the vectors anticlockwise about each axis, assuming that therotation axis considered is pointing towards the reader.

C.2 Axioms of vector addition and scalar multiplication

Vectors can undergo algebraic operations such as addition and scalar multiplica-tion. The axioms that define these operations in linear algebra are summarised inTable C.1.

Table C.1 Axiomatic properties of vector addition and scalar multiplication.

Axiomatic property Vector expression

associationv1 + (v2 + v3) =

(v1 + v2) + v3

commutation v1 + v2 = v2 + v1

sum identity(existence of nullvector)

v + 0 = v

inversion (existenceof –v)

v + (–v) = 0

distributionscalar times vector sum s((v1 + v2) = sv1 + sv2

scalar sum times vector (s1 + s2)v = s1v + s2v

scalar times vector s1(s2v) = (s1s2)v

multiplicationidentity (existence ofs = 1)

1(v) = v


C.3 Vector multiplication

C.3.1 Scalar product

The projection of some vector A onto its Cartesian coordinates provides a special caseof the definition of the scalar or ‘dot’ product from which we can get a general definitionbetween two arbitrary vectors A and B. As shown in FigureC.3 we project A onto thecoordinate axes x, y, z and define this projection as the ‘dot’ product of A and the unitvectors x, y, z:

Ax = A cosα ≡ A · xAy = A cosβ ≡ A · yAz = A cos γ ≡ A · z

We posit that the scalar product is associative and distributive:

A · (yB) = (yA) · B = yA · B (C.11)

A · (B +C) = A · B +A · B (C.12)

We use the associative property to get a generalisation of the dot product between twovectors A and B. We can write B in terms of its components and the unit vectors alongthe Cartesian axes:

B = Bxx + Byy + Bzz

Then,

A · B = A · (Bxx + Byy + Bzz)

(C.13)

But then invoking the associative property,

A · B = BxA · x + ByA · y + BzA · z= BxAx + ByAy + BzAz

θ

A

Ax Bx

B

x

y

Figure C.3 Vector scalar product. A · B =AB cos θ .


Therefore:

A · B =∑i

BiAi =∑i

AiBi = B · A (C.14)

and we see that the scalar product between two vectors is commutative as well as asso-ciative. In the same way we defined Ax = A · x = A cosα, we could define the projectionof A onto the B direction as

AB = A cos θ ≡ A · B = A · B/B

Then multiplying this last equivalency by B,

BA cos θ = A · B

Similarly, we could specify the projection of B onto the A direction. Then BA =B cos θ ≡ B · A/A. Then

AB cos θ = B · A

Once again we see that the scalar product between any two vectors is commutative andequal to the product of the two vector ‘lengths’ and the cosine of the angle between them.

The distributive property, EquationC.12, is illustrated in FigureC.4. The figureshows geometrically that the sum of the projections of B and C onto A is equivalentto the projection of their vector sum B +C onto A:

B · A +C · A = (B +C) · A

C.3.2 Vector product

The vector or ‘cross’ product of two vectors A and B has meaning in 3-D and isdefined as

A×B = C (C.15)

The vector product of A and B is another vector C. The magnitude of C is related tothose of A and B by

B

C

A

B + C

Figure C.4 Distributive law of vectoraddition. A · (B + C) = A ·B + A · C.


|C| = |A| · |B| sin θ (C.16)

where θ is the angle between vectors A and B. The direction of C is governed by the‘right-hand screw rule’. The vectors A and B, separated by angle θ , form a plane. AsA rotates towards B, the positive direction of C points in the same direction as would aright-hand threaded screw. A consequence of this convention is that

A×B = –B×A (C.17)

The cross product is anticommutative. From the definition we can directly obtain basicrelations among the Cartesian unit vectors:

x× y = z y× z = x z× x = y (C.18)

and

y× x = –z z× y = –x x× z = –y (C.19)

It is also clear from the definition, EquationC.16 that

x× x = y× y = z× z = 0 (C.20)

Notice that these relations can be generated by a cyclic commutation of x, y, z startingfrom any one of them. In analogy to the scalar product we posit that the cross productoperation is distributive and associative. Thus:

A× (B +C) = A×B +A×C (C.21)

(A +B)×C = A×C +B×C (C.22)

A× (sB) = sA×B = (sA)×B (C.23)

Now let us take A×B and decompose this expression into its Cartesian components:

A×B =(Axx + Ayy + Azz

)× (Bxx + Byy + Bzz)

(C.24)

= (AyBz – AzBy)x – (AxBz – AzBx)y +(AxBy – AyBx

)z

= Cxx +Cyy +Czz = C (C.25)

We see that the Cartesian components of C are constructed from combining products ofthe Cartesian components of A and B. Note also that these combinations commute cyc-lically. The easiest way to generate the components of the vector product is by writing adeterminant and expanding it. The first row of the determinant consists of the Cartesianunit vectors, x, y, z. The Cartesian components of A composes the second row and theCartesian components of B, the third:


A

BC

Figure C.5 Geometrical interpretation of the triplescalar product, A · (B× C).

C =

∣∣∣∣∣∣x y zAx Ay AzBx By Bz

∣∣∣∣∣∣ (C.26)

Expansion of this determinant results in an expression identical to the second line ofEquationC.24, obtained from the cross product definition applied to the Cartesian unitvectors.

C.3.3 Triple products

Because the cross product of two vectors is itself a vector we can ‘dot’ this result and‘cross’ it with yet another vector. Since the dot operation results in a scalar quantity, thistriple product is called the scalar triple product. Figure C.5 shows that the scalar tripleproduct can be interpreted geometrically as the volume of a parallelepiped. Furthermore,the scalar triple product is cyclically commutative:

A · (B×C) = B · (C×A) = C · (A×B) (C.27)

Because of the anticommutivity of the cross product operation,

A · (B×C) = –A · (C×B)

The vector triple product can be simplified by the ‘BAC-CAB’ rule:

A× (B×C) = B(A · C) –C(A · B) (C.28)

which can be verified by direct expansion in Cartesian coordinates.

C.4 Vector fields

The idea of a field is the assignment of some value or function to all points in a regionof geometric space. If the entity assigned is a scalar, then we have a scalar field. Thedistribution of temperature throughout the solar system is an example of a scalar field.In the theory of electromagnetism, the concept of a vector field is essential. A vector


field is an assignment of a vector to each point in some region of geometrical space.The electric E and magnetic B force fields produced by an oscillating dipole source orpropagating in a plane wave are examples of vector force fields.

Associated with vector and scalar fields are three differential operations important inphysics and engineering: the gradient, the divergence, and (in 3-D) the curl operations.

C.4.1 Gradient

If ϕ(x, y, z) is a scalar field then the gradient of ϕ is defined as

∇ϕ(x, y, z) = ∂ϕ

∂xx +

∂ϕ

∂yy +

∂ϕ

∂zz (C.29)

Notice that the gradient of scalar field is a vector field consisting of the partial deriva-tives of the scalar field pointing in the x, y, z directions. If we just considered the totaldifferential of ϕ in terms of its partial derivatives we have

dϕ =∂ϕ

∂xdx +

∂ϕ

∂ydy +

∂ϕ

∂zdz (C.30)

In fact, we can write this scalar differential as the scalar product of ∇ϕ and the vectorcoordinate differential dr = dx x + dy y + dz z:

dϕ = ∇ϕ · dr = |∇ϕ||dr| cos θ (C.31)

where the angle θ is the angle between the gradient and the coordinate vector. Clearly,in order for EquationC.31 to be in accord with EquationC.30, the angle θ = 0. Ifθ were any other value, then the dot product in EquationC.31 would be less thandϕ. So ∇ϕ as defined in EquationC.29 points in the direction of maximum changein ϕ and its magnitude |∇ϕ| yields the greatest rate of change (slope) along thisdirection.

The ∇ = ∂/∂x x + ∂/∂y y + ∂/∂z z is a vector operator which acts on a scalar field toproduce a vector field. The most familiar example of the use of the gradient in elec-tromagnetism is the relation between an electrostatic field E and an electrical potentialV (x, y, z):

E = –∇V

The gradient operator is sometimes called ‘grad’ or ‘del’ and the gradient operation ona scalar field ϕ is termed ‘gradϕ’ or ‘delϕ’. The ∇ symbol itself is called the ‘nabla’.

C.4.2 Divergence

The divergence operation is the dot product of the vector gradient operator with a vectorfield. Suppose we have a vector field ϕ. The result of the divergence operation is


∇ · ϕ =[∂

∂xx +

∂

∂yy +

∂

∂zz]

· [ϕxx + ϕyy + ϕzz]

=∂ϕx

∂x+∂ϕy

∂y+∂ϕz

∂z(C.32)

Notice that the divergence operation results in a scalar field because it is a ‘dot prod-uct’ operation between the vector operator ∇ and the vector field ϕ. The divergence ofa vector field is a measure of how much the field increases or diverges along the fieldcoordinates. A vector field radiating from a point in all directions has a very high di-vergence. A vector field whose magnitude is invariant along some coordinate has a nulldivergence. A positive (negative) divergence denotes an increasing (decreasing) field.This geometrical representation of divergence is shown in FigureC.6. The divergenceof vector field is sometimes written as divϕ. The most familiar example of the use of thedivergence in electromagnetism is the differential form of Gauss’s law,

∇ · E =ρ

ε0(C.33)

where E is the electric field and ρ the electric charge density.

C.4.3 Curl

The vector product of ∇ and a vector field ϕ is another vector field χ , called the curlof ϕ:

χ = ∇ × ϕ (C.34)

(a) (b)

x

x

y y

Figure C.6 Panel (a) shows a vector field with very high negative divergence. Vector amplitudesdecrease with distance from origin. Panel (b) shows a constant vector field along x which exhibits nulldivergence.


From the determinant rule for the cross product of two vectors we can write

χ =

∣∣∣∣∣∣∣x y z∂∂x

∂∂y

∂∂z

ϕx ϕy ϕz

∣∣∣∣∣∣∣ (C.35)

Expanding the determinant, we have explicitly,

χ =[∂ϕz

∂y–∂ϕz

∂y

]x –

[∂ϕz

∂x–∂ϕx

∂z

]y +

[∂ϕy

∂x–∂ϕx

∂y

]z (C.36)

The curl of a vector field is measure of its ‘rotation’ around an axis perpendicular tothe plane of rotation. A vortex field has a very large curl while a field emanating radiallyfrom a point in all directions has no curl. Figure C.7 shows a field of all curl and nulldivergence. The explicit expression for this vector field is

V(x, y) = –y√

x2 + y2i +

x√x2 + y2

j (C.37)

Figure C.8 shows a vector field with finite divergence but null curl along the x = ydiagonals. The expression for this vector field is

V(x, y) =y√

x2 + y2i +

x√x2 + y2

j (C.38)

One can show that the divergence of the curl of a vector field is null:

∇ · [∇ × (∇ϕ)] = 0 (C.39)

and the curl of the gradient of a scalar field is null:

∇ × (∇ϕ) = 0 (C.40)

X

IYZ

Figure C.7 A vector field circulating anti-clockwise in the x–y plane producing a curl fieldalong z but no divergence.


X

Y

Figure C.8 A vector field with a null curl atthe origin and along the x = y diagonals butwith finite divergence for all x, y except at thepoint x = 0, y = 0 where the divergence is null.

The most celebrated curl equation in electromagnetism is Faraday’s law:

∇ × E = –∂B∂t

In this discussion we have described the grad, div, and curl operations in termsof Cartesian coordinates. But many other curvilinear coordinate systems are a moreconvenient choice, depending on the symmetry of the problem. Two commonly encoun-tered curvilinear coordinate systems in physics are cylindrical and polar coordinates. Theexpressions for grad, div, and curl in these coordinates is discussed in Appendix D.

C.5 Integral theorems for vector fields

C.5.1 Integral of the gradient

We saw in EquationC.31 that the differential of a scalar field dϕ is equal to the dotproduct of the gradient of the field ∇ϕ and the differential of coordinate length dr. If weintegrate the differential dϕ from ϕ(x0, y0, z0) to ϕ(x1, y1, z1) we have

∫ 1

0dϕ = ϕ(x1, y1, z1) – ϕ(x0, y0, z0) =

∫ 1

0∇ϕ · dr (C.41)

Equation C.41 says that the line integral of the vector field ∇ϕ using the path dr isequal to the difference between the scalar field values at the two end points. Since thisdifference does not depend on the path, the result of the line integral is independent ofthe path taken; and in particular,

∮∇ϕ · dr = 0 (C.42)


for any closed loop since for these the ending point is identical to the starting point.The integral over a path of the gradient of a scalar field is commonly encountered inelectrostatic problems where the task is to calculate the potential difference between,say, two conductors that form an electric field between them. The potential difference isgiven by

V21 = V2 – V1 = –∫ 2

1E · dr = –

∫ 2

1∇V · dr (C.43)

C.5.2 Integral of the divergence

The expression for the divergence theorem or Gauss’s theorem is

∫vol(∇ · χ) dV =

∮surf

χ · dS (C.44)

where dV is the differential volume element and dS is a differential element of the sur-face surrounding (bounding) the volume. Here χ is a vector field, in contrast to ϕ ofEquation C.31, a scalar field. Note that the surface element dS is itself a differentialvector with the unit vector direction perpendicular to the surface and conventionallypointing outwards. A common application of the divergence theorem is the integral formof Gauss’s law, discussed in Chapter 2, Equation 2.124:

∫vol

∇ ·D dV = ε0

∫vol

∇ · E dV =∫volρ dV = ε0

∫surf

E · dS

where D is the displacement vector field and E is the electric field emanating from thecharge density ρ enclosed by the surface boundary S of volume V . Note that the surfacedifferential element is a vector differential with direction pointing outwards from theboundary and perpendicular to it.

C.5.3 Integral of the curl

The surface integral over the curl of a vector is called Stokes’ theorem:

∫surf

(∇ × χ) · dS =∮

χ · dr (C.45)

The line integral on the right is taken around the boundary of the surface integral onthe left. A convenient way to remember and visualise the geometric interpretation of thecurl integral is to think of Ampère’s law. In differential form it is

∇ ×H = Jfree


whereH is the magnetic field and Jfree is a free charge current density. The integral formof this equation is, taking into account the curl theorem,∫

surf(∇ ×H) · dS =

∮H · dr =

∫surf

Jfree · dS = I (C.46)

where I is the charge current running in a wire of cross section S. This relation can bepictured as a loop of H-field surrounding a straight wire and generated by a current Irunning through it.

C.6 Useful identities of vector calculus

a · b× c = b · c× a – c · a× b (C.47)

a× (b× c) = (a · c)b – (a · b)c (C.48)

(a× b) · (c× d) = a · b× (c× d) (C.49)

= a · (b · d c – b · c d) (C.50)

= (a · c)(b · d) – (a · d)(b · c) (C.51)

(a× b)× (c× d) = (a× b · d)c – (a× b · c)d (C.52)

∇(ϕ + ψ) = ∇ϕ + ∇ψ (C.53)

∇(ϕψ) = ϕ∇ψ + ψ∇ϕ (C.54)

∇ · (a + b) = ∇ · a + ∇ · b (C.55)

∇ × (a + b) = ∇ × a + ∇ × b (C.56)

∇ · (ϕa) = a ·∇ϕ + ϕ∇ · a (C.57)

∇ × (ϕa) = ∇ϕ × a + ϕ∇ × a (C.58)

∇(a · b) = (a ·∇)b + (b ·∇)a + a× (∇ × b) + b× (∇ × a) (C.59)

∇ · (a× b) = b ·∇ × a – a ·∇ × b (C.60)

∇ × (a× b) = a∇ · b – b∇ · a + (b ·∇)a – (a ·∇)b (C.61)

∇ ×∇ × a = ∇∇ · a – ∇2a (C.62)

∇ ×∇ϕ = 0 (C.63)

∇ ·∇ × a = 0 (C.64)

Appendix DGradient, Divergence, and Curlin Cylindrical and Polar Coordinates

Although Cartesian coordinates are the most familiar and serve many purposes, theyare not the only orthogonal coordinate system that can be used to define a set of basisvectors. Often we can take advantage of the natural symmetry of a problem to expressvectors in some other orthogonal curvilinear coordinate system. The most common ofthese are the cylindrical and polar coordinates because they are appropriate for manypractical problems.

In general we can expand a vector V in basis vectors of the Cartesian system or someother system with basis vectors {q},

V = xVx + yVy + zVz or V = q1V1 + q2V2 + q3V3 (D.1)

However, we cannot treat the position vector with this general rule. In this special case

r = xx + yy + zz cartesian coordinates (D.2)

r = ρρ + zz cylindrical coordinates (D.3)

r = rr spherical coordinates (D.4)

In the general case we convert from the Cartesian system to another through the relations

x = x(q1, q2, q3)

y = y(q1, q2, q3)

z = z(q1, q2, q3)

The complete differential in x can be written as

dx =∂x∂q1

dq1 +∂y∂q2

dq2 +∂y∂q3

dq3

and similarly for y and z. The differential of the position vector r in the Cartesian basis is

dr = dx + dy + dz

Appendix D: Gradient, Divergence, and Curl in Cylindrical and Polar Coordinates 389

and the square of the distance between two neighbouring points can then be written as

ds2 = dr · dr = dx2 + dy2 + dz2

But the differential of dr can also be expanded in the curvilinear coordinate system:

dr =∂r∂q1

dq1 +∂r∂q2

dq2 +∂r∂q3

dq3 (D.5)

and

ds2 = dr · dr =∑ij

∂r∂qi· ∂r∂qj

dqidqj (D.6)

Remembering that we have supposed that the curvilinear coordinates are orthogonal,

qi · qj = δijwe write EquationD.6:

ds2 =∑i

∂2r∂q2i

dq2i = g11dq21 + g22dq22 + g33dq

23 =

∑i

(hidqi)2

where we have identified gii = h2i . We see from EquationD.6 that hi is a scale factor suchthat the differential length segment dsi can be written as

dsi = hidqi and∂r∂qi

= hiqi

and from EquationD.5 the position vector differential dr can be expanded in terms ofthese scale factors in the curvilinear basis as

dr = h1dq1q1 + h2dq2q2 + h3dq3q3 (D.7)

From the line segment dsi we can easily write the surface and volume elements σ , τ inthe curvilinear system:

dσij = dsidsj = hihj dqidqj i �= j

and

dτ = ds1ds2ds3 = h1h2h3 dq1dq2d3

We can expand an area vector σ in the curvilinear basis set analogous to the positionvector r in EquationD.7:

390 Appendix D: Gradient, Divergence, and Curl in Cylindrical and Polar Coordinates

dσ = ds2ds3dq1 + ds3ds1q2 + ds1ds2q3= h2h3 dq2dq3q1 + h3h1 dq3dq1q2 + h1h2 dq1dq2q3 (D.8)

For example, we can take an ordinary vector quantity F and expand it in Cartesiancoordinates or in spherical coordinates:

F = Fxx + Fyy + Fzz (D.9)

F = Fr r + Fθ θ + Fϕϕ (D.10)

The unit vectors r, θ , ϕ in terms of the Cartesian unit vectors are

r = sin θ cosϕ x + sin θ sinϕ y + cos θ z (D.11)

θ = cos θ cosϕ x + cos θ sinϕ y – sin θ z (D.12)

ϕ = – sinϕ x + cosϕ y (D.13)

The differential length vector along the surface in spherical coordinates is then

dl = dr r + r dθ θ + r sin θ dϕ ϕ (D.14)

and the differential volume element in spherical coordinates is obtained from the productof ds1 ds2 ds3 with

ds1 = h1 dq1 = dr (D.15)

ds2 = h2 dq2 = r dθ (D.16)

ds3 = h3 dq3 = r sin θ dϕ (D.17)

The differential volume element in spherical coordinates is therefore:

dV = ds1 ds2 ds3 = dr rdθ r sin θdϕ = r2 sin θ dθ dϕ dr (D.18)

The expressions for dr and dσ in EquationsD.7 and D.8 give us the tools to write vectorline and surface integrals in the curvilinear coordinates:

∫V · dr =

∑i

∫Vihidqi (D.19)

∫V · dσ =

∑i

∫Vihjhk dqjdqk j �= k �= i (D.20)


D.1 The gradient in curvilinear coordinates

The gradient is a vector operator ∇ that operates on a scalar point function ψ . InCartesian coordinates we write

∇ψ =(∂

∂xx +

∂

∂yy +

∂

∂zz)ψ(x, y, z) (D.21)

An alternative integral definition is more convenient for finding the expression for thegradient in some curvilinear coordinate system:

∇ψ = lim∫dτ→0

∫ψdσ∫dτ

(D.22)

The gradient, in general, is defined so that it yields the maximum spatial rate of changeof the scalar function, and this maximum spatial change is independent of the coordinatesystem in which it is described. Therefore, we should be able to express it in coordin-ate systems other than Cartesian. Remembering that the length segments dx, dy, dz inCartesian coordinates go over into ds1, ds2, ds3 in our chosen curvilinear system, we canwrite the gradient in our new coordinates as

∇ψ(q1, q2, q3) = q1∂ψ

∂s1+ q2

∂ψ

∂s2+ q3

∂ψ

∂s3(D.23)

and also remembering that dsi = hidqi we have

∇ψ(q1, q2, q3) = q11h1

∂ψ

∂q1+ q2

1h2

∂ψ

∂q2+ q3

1h3

∂ψ

∂q3(D.24)

D.2 The divergence in curvilinear coordinates

The divergence ∇ · V of some vector field V can be expressed as a differential inCartesian coordinates as(

∂

∂xx +

∂

∂yy +

∂

∂zz)· (Vxx + Vyy + Vzz

)(D.25)

Similar to the gradient operation, the divergence can be generalised to curvilinear co-ordinates by considering it as the result of taking the limit of a differential vector fieldsurface integral divided by the differential volume enclosed by the surface as the volumeapproaches zero. Taking this limiting ratio operation as the definition of the divergencewe have

∇ · V (q1, q2, q3) = lim∫dτ→0

δ∫V · dσ∫dτ


and the differential of the surface integral is

δ

∫V (q1, q2, q3) · dσ =[

∂(V1h2h3)∂q1

+∂(V2h3h1)

∂q2+∂(V3h1h2)

∂q3

]dq1dq3dq3

and taking∫dτ in the limit as ds1ds2ds3 = h1dq1 h2dq2 h3dq3 we have for the limiting ratio

∇ · V (q1, q2, q3) =1

h1h2h3

[∂

∂q1(V1h2h3) +

∂

∂q2(V2h3h1)+

∂

∂q3(V3h1h2)

](D.26)

Another useful general expression is the scalar Laplacian operator, ∇ ·∇, which we canget from combining the grad and div operations,

∇ ·∇ψ =1

h1h2h3

[∂

∂q1

(h2h3h1

∂ψ

∂q1

)+

∂

∂q2

(h3h1h2

∂ψ

∂q2

)+

∂

∂q3

(h1h2h3

∂ψ

∂q3

)](D.27)

D.3 The curl in curvilinear coordinates

The familiar differential expression for the curl operation in Cartesian coordinates is

∇ × V =(∂Vz∂y

–∂Vy∂z

)x +

(∂Vx∂z

–∂Vz∂x

)y +

(∂Vy∂x

–∂Vx∂y

)z

and the easiest way to remember it, is by writing the expression as determinant:

∇ × V =

∣∣∣∣∣∣∣x y z∂∂x

∂∂y

∂∂z

Vx Vy Vz

∣∣∣∣∣∣∣As with the gradient and divergence operations, the curl can also be written as thelimiting integral operation,

∇ × V = lim∫dτ→0

∫dσ × V∫dτ

It can be shown that by using this integral limiting form, and applying Stokes’ theorem,the curl operation can be written in curvilinear coordinates as


∇ × V =1

h1h2h3

∣∣∣∣∣∣∣q1h1 q2h2 q3h3∂∂q1

∂∂q2

∂∂q3

h1V1 h2V2 h3V3

∣∣∣∣∣∣∣ (D.28)

D.4 Expressions for grad, div, curl in cylindrical and polar coordinates

D.4.1 Circular cylindrical coordinates

Circular cylindrical coordinates consist of three independent variables, ρ, ϕ, z and theirassociated unit vectors, ρ, ϕ, z. The limits on these variables are,

0 ≤ ρ ≤ ∞ 0 ≤ ϕ ≤ 2π –∞ < z < +∞

Figure D.1 shows the relation between the Cartesian coordinates x, y, z and the cylin-drical coordinates ρ,ϕ, z. The z coordinate is common to both systems. From FigureD.1it is evident that

ρ =√(x2 + y2

)x = ρ cosϕ y = ρ sinϕ z = z

The differential volume element dV is

dV = dsρ dsϕ dsz = ρdρ ρdϕ dz

y

x

x

φρ

φ

z

y

z z

Figure D.1 Cylindrical coordinates ρ,ϕ, z and unit vectorsρ, ϕ, z and their relations to Cartesian coordinates and unitvectors, x, y, z, x, y, z.


From which we identify

h1 = hρ = 1 h2 = hϕ = ρ h3 = hz = 1

Then from the general expressions for grad, div, curl, EquationsD.24,D.26, and D.28we write these operators in cylindrical coordinates as

grad ∇ψ(ρ,ϕ, z) = ∂ψ

∂ρρ +

1ρ

∂ψ

∂ϕϕ +

∂ψ

∂zz (D.29)

div ∇ · V =1ρ

∂(ρVρ

)∂ρ

+1ρ

∂Vϕ∂ϕ

+∂Vz∂z

(D.30)

curl ∇ × V =1ρ

∣∣∣∣∣∣∣ρ ρϕ z∂∂ρ

∂∂ϕ

∂∂z

Vρ ρVϕ Vz

∣∣∣∣∣∣∣ (D.31)

We can also find the scalar Laplacian in cylindrical coordinates by applying the generalexpression EquationD.27:

Laplacian ∇2ψ =1ρ

∂

∂ρ

(ρ∂ψ

∂ρ

)+

1ρ2

∂2ψ

∂ϕ2+∂2ψ

∂z2(D.32)

D.4.2 Polar coordinates

Polar coordinates consist of three independent variables r, θ ϕ and their associated unitvectors r, θ , ϕ. The limits on these variables are,

0 ≤ r ≤ ∞ 0 ≤ θ ≤ π 0 ≤ ϕ ≤ 2π

x

y

z

φ

x

y

r

z

θθ

φ

Figure D.2 Polar coordinates r, θ ,ϕ andunit vectors r, θ , ϕ and their relations toCartesian coordinates and unit vectors,x, y, z, x, y, z.


Figure D.2 shows the relation between the Cartesian coordinates x, y, z and the polarcoordinates r, θ ,ϕ. From FigureD.2 we see that

r =√x2 + y2 + z2 x = r sin θ cosϕ y = r sin θ cosϕ z = r cos θ

From the differential volume dV and inspection of FigureD.2 we can identify the scalefactors h1, h2, h3:

dV = dsr dsθ dsϕ = dr rdθ r sin θdϕ

hr = 1 hθ = r hϕ = r sin θ

Then from EquationsD.24,D.26, and D.28 we write expressions for the grad, div, andcurl operators in polar coordinates:

grad ∇ψ =∂ψ

∂rr +

1r∂ψ

∂θθ +

1r sin θ

∂ψ

∂ϕϕ (D.33)

div ∇ · V =1

r2 sin θ

[sin θ

∂r2Vr∂r

+ r∂ sin θVθ∂θ

+ r∂Vϕ∂ϕ

](D.34)

curl ∇ × V =1

r2 sin θ

∣∣∣∣∣∣∣r rθ r sin θ ϕ∂∂r

∂∂θ

∂∂ϕ

Vr rVθ r sin θVϕ

∣∣∣∣∣∣∣ (D.35)

Finally, again by applying EquationD.27 we can obtain the Laplacian operator in polarcoordinates:

Laplacian ∇2ψ =1

r2 sin θ

[sin θ

∂

∂r

(r2∂ψ

∂r

)+∂

∂θ

(sin θ

∂ψ

∂θ

)+

1sin θ

∂2ψ

∂ϕ2

](D.36)

Or, expanding the radial terms,

Laplacian ∇2ψ =∂2ψ

∂r2+

2r∂ψ

∂r+

1r2 sin θ

[∂

∂θ

(sin θ

∂ψ

∂θ

)+

1sin θ

∂2ψ

∂ϕ2

](D.37)

Appendix EProperties of Phasors

E.1 Introduction

In Chapter 3, Section 3.1.1, we discussed the phasor form for electromagnetic propa-gating fields consisting of a single-frequency harmonic time-dependence factor e–iωt andthe spatially dependent amplitude-modulation factor ei(k·r+ϕ). The general form of thefield is

A(r, t) = A0ei(k·r+ϕ)e–iωt (E.1)

and we separated the field into a phasor factor, involving only the spatial dependence,and a harmonic time factor that was usually invariant.

In general, a ‘phasor’ is any function with a sinusoidal modulation, and in thisAppendix, we examine some of the useful properties of phasors and their applicationto circuit theory.

E.1.1 Phasor addition

Since phasors can be represented as complex exponentials, multiplication of two phasorsAeiα and Beiβ is elementary:

Aeiα · Beiβ = ABei(α+β) = Ceiγ (E.2)

The product of two phasors is another phasor with a phase angle equal to the algebraicsum of the two individual arguments of the exponential factors.

The addition of two phasors is less obvious. In many cases we are interested in realsin and cos functions so let us examine the following sum of two real phasors, V1(t) =A sinωt and V2(t) = B sin(ωt + ϕ):

V3(t) = V1(t) + V2(t)

= A sinωt + B sin(ωt + ϕ) (E.3)

On intuition we might suppose that the sum of two sin functions differing only in amp-litude and relative phase would be another sin function. We posit therefore that the formof the sum is

V3(t) = C sin(ωt + δ) (E.4)

Appendix E: Properties of Phasors 397

x

y

A

CB

φ

φ

γ

Figure E.1 Relations among two phasors A and Bthat differ in amplitude and phase. Phase of A leadsphase of B by ϕ. The interior angle γ is related to ϕby γ = π – ϕ.

where C and δ are to be determined. Expanding V3(t) in Equations E.3 and E.4, andsetting coefficients of sinωt and cosωt equal, we find that

δ = tan–1[

B sinϕA + B cosϕ

](E.5)

C =[A2 + B2 + 2AB cosϕ

]1/2(E.6)

The expression forC, the amplitude of the phasor sum, is highly reminiscent of the resultwe would expect for the length of a vector C that is the vector sum of A and B. In thiscase the length of the vector C would be determined by the law of cosines as indicatedin Figure E.1. The interior angle γ between the two component vectors is closely relatedto ϕ, the relative angle between the two phasors. In fact, it is easy to show that

γ = π – ϕ (E.7)

and substitution into EquationE.6 results in

C =[A2 + B2 – 2AB cos γ

]1/2(E.8)

confirming that two phasors add with amplitude equivalent to the resultant ‘length’ fromthe sum of two vectors.

E.2 Application of phasors to circuit analysis

As an illustration of the usefulness of phasors to harmonic circuit analysis, we considera simple RLC circuit with

V (t) = Ri(t) + Ldi(t)dt

+1C

∫i(t)dt (E.9)

where V ,R,L, and C have their usual meanings of voltage, resistance, inductance, andcapacitance, respectively, and i(t) is the time-dependent current running through all the

398 Appendix E: Properties of Phasors

lumped circuit elements. We posit that the current source is harmonically oscillating atfrequency ω and write i(t) as a phasor,

i(t) = i0e–iωt (E.10)

Substituting the phasor form into EquationE.9 we find that

V (t) = Ri0e–iωt + ωLi0e–i(ωt+π /2) +1ωC

i0e–(ωt–π /2) (E.11)

= VR + VL + VC (E.12)

We see that the voltage drop across the resistor is in phase with the source, while theinductive reactance, ωL, shows a phase advance of π /2 and the capacitive reactance, 1/ωC,lags in phase by π /2. The resistive voltage drop is purely dissipative while the two react-ances store energy originating at the source and return it to the circuit at different pointsalong the harmonic cycle. Equation E.9 can be rewritten in terms of the impedance, Z(ω),

v(t)

v(t)

v(t)i(t)

VR(t)

VR (t)

iR(t) iL(t)

i(t)

iC(t)iC(t)

iR(t)

iL(t)v(t)

R

R L C

(a) (b)

(c) (d)

L C

VL(t)

VL (t)

VC (t)

i(t)

VC(t)

ϕ

ϕ

Figure E.2 (a) Series RLC circuit with harmonic voltage source and a commoncurrent i(t) running through each circuit element. (b) Voltage phasor diagram showingthe phase ‘lead’ for the capacitive voltage term, the phase ‘lag’ for the inductive voltageterm, and the net phase difference between the driving voltage and the resistive voltagedrop. Panels (c) and (d) show similar diagrams for current phasors in RLC parallelcircuit.

Appendix E: Properties of Phasors 399

V (t) = Z(ω)i0e–iωt (E.13)

Z(ω) = R + (ωL)e–i(π /2) +(

1ωC

)ei(π /2) (E.14)

= R – iωL +(

iωC

)(E.15)

= R + iXL(ω) + iXC(ω) (E.16)

where XL and XC are the inductive and capacitive reactances in phasor form. We cantake this analysis further by considering the RLC circuit in series and in parallel. FigureE.2 shows circuit and phasor diagrams for the two cases. In the series circuit, the prop-erty common to all lumped elements is the current i(t) so the relevant phasor quantitiesare the voltages across the resistive, inductive, and capacitive elements. For the paral-lel circuit, the common property is the voltage V (t) and the relevant phasor diagram isexpressed in terms of the individual currents passing through each circuit element.

Appendix FProperties of the Laguerre Functions

The Laguerre polynomials and the ‘associated’ Laguerre polynomials are solutions toLaguerre’s differential equation and are closely related to the radial solutions of theSchrödinger equation for the hydrogen atom.

F.1 Generating function and recursion relations

The generating function for the Laguerre polynomials is given by

g(x, z) =e–xz/(1–z)

1 – z=∞∑n=0

Ln(x)zn |z| ≤ 1 (F.1)

As we did with Legendre and Hermite polynomials, we find recurrence relations betweenadjacent members of the Laguerre polynomials by differentiating the generating functionwith respect to z and x:

(1 – z2)∂g(x, z)∂z

= (1 – x – z)g(x, z) (F.2)

(z – 1)∂g(x, z)∂x

= zg(x, z) (F.3)

Substituting EquationF.1 into EquationF.2, adjusting the summation indices, andequating terms with equal powers of zn, results in the first recurrence relation:

Ln+1(x) =(2n + 1 – xn + 1

)Ln – nLn–1 (F.4)

It is often useful to have a recurrence relation involving the derivatives of the poly-nomials, and we can obtain one by using Equations F.2 and F.3 to derive a thirdrelation,

x∂g(x, z)∂x

= z∂g(x, z)∂z

– z∂[zg(x, z)]

∂z(F.5)

Appendix F: Properties of the Laguerre Functions 401

Making the appropriate substitutions from EquationF.1, adjusting summation indices,and equating terms with like powers of zn results in

xL′n(x) = nLn(x) – nLn–1(x) (F.6)

which is the second recursion relation for Laguerre polynomials.

F.2 Orthogonality and normalisation

The Laguerre polynomials themselves do not form an orthogonal set of functions, butthey can be made orthogonal by joining a factor e–x to them. Thus:∫ ∞

0e–xLm(x)Ln(x) = δmn (F.7)

where the polynomials defined by the generating function, EquationF.1 are alreadynormalised. As in the case of Hermite polynomials, we can identify a closely relatedLaguerre function ϕn(x) that has the orthonormal properties of a complete set offunctions involving the Laguerre polynomials,

ϕn(x) = e–x/2Ln(x) (F.8)

The ordinary differential equation that ϕn(x) satisfies, however, is not the physically rele-vant one with which we associate the Schrödinger radial equation for the hydrogen atom.To find the solutions for that equation we need to examine the ‘associated’ Laguerrepolynomials.

F.3 Associated Laguerre polynomials

The associated Laguerre polynomials are defined in terms of the Laguerre polyno-mials by,

Lkn(x) = (–1)kdk

dxkLn+k(x) (F.9)

and given the Rodrigues form for the Laguerre polynomials,

Ln(x) =ex

n!dn

dxn(xne–x) (F.10)

from which, after writing the Taylor series expansion of the exponential terms, we canwrite a series representation for the ‘normal’ Laguerre polynomials with integral n as

Ln(x) =n∑

m=0

(–1)mn!xm

(n –m)!m!m!(F.11)

402 Appendix F: Properties of the Laguerre Functions

We can obtain a similar series representation for the associated Laguerre polynomials bysubstituting EquationF.11 into the definition, EquationF.9:

Lkn(x) =n∑

m=0

(–1)m(n + k)!

(n –m)!(k +m)!m!xm k > –1 (F.12)

The first few associated Laguerre polynomials are, therefore,

Lk0(x) = 1

Lk1(x) = –x + k + 1

Lk2(x) =x2

2– (k + 2)x +

(k + 2)(k + 1)2

F.3.1 Generating function, recurrence relations, and orthonormality

The generating function for the associated Laguerre polynomials is given by

e–xz/(1–z)

(1 – z)k+1=∞∑n=0

Lkn(x)zn |z| < 1 (F.13)

from which recurrence relations can be derived by differentiation and equating of termswith like powers in z in the usual way. The resulting recurrence relation is

(n + 1)Lkn+1(x) = (2n + k + 1 – x)Lkn(x) – (n + k)Lkn–1(x) (F.14)

The associated Laguerre polynomials are solutions to the associated Laguerre ordinarydifferential equation, but this equation is not ‘self-adjoint’ or hermitian, and thereforecannot function as a legitimate quantum mechanical operator. However, it can be castinto a form suitable for quantum mechanics by multiplying the associated Laguerrepolynomials by a factor, e–x/2xk/2Lkn(x). This factor transforms the polynomials into theassociated Laguerre functions:

ϕkn(x) = e–x/2xk/2Lkn(x) (F.15)

These associated Laguerre functions have the desired orthonormality,∫ ∞0

e–xxkLkn(x)Lkm(x) dx =

(n + k)!n!

δmn (F.16)

and they satisfy the ‘quantum mechanically correct’ ordinary differential equation,

xd2ϕkn(x)dx2

+dϕkn(x)dx

+(–x4+

2n + k + 12

–k2

4x

)ϕkn(x) = 0 (F.17)

Appendix F: Properties of the Laguerre Functions 403

This equation is ‘correct’ in the sense that the operator integrals corresponding to aver-age values of observable variables, and the eigenvalues themselves, are guaranteed tobe real. But in fact, EquationF.17 is still not the ordinary differential equation corres-ponding to the form of the radial Schrödinger equation for the hydrogen atom. Theproblem is that the Schrödinger equation does not exhibit a term with a first-derivativeoperator. However, this term can be eliminated by multiplying the Laguerre function bya factor x1/2:

kn(x) = e–x/2x(k+1)/2Lkn(x) (F.18)

This family of modified Laguerre functions is a solution to the ordinary differentialequation,

d2kn(x)

dx2+(–14+

2n + k + 12x

–k2 – 14x2

)kn(x) = 0 (F.19)

and has a normalisation:∫ ∞0

e–xxk+1[Lkn(x)

]2dx =

(n + k)!n!

(2n + k + 1) (F.20)

Notice that in EquationF.19 there is no term in the first derivative of x, and thereforethis second-order differential equation and its family of modified Laguerre functions canserve as a prototype for the radial Schrödinger equation of the one-electron atom.

Appendix GProperties of the Legendre Functions

The Legendre functions are a family of solutions to the Legendre differential equation.This equation determines the angular behaviour of many physical problems includingthe scalar Helmholtz wave equation in optics, Maxwell’s equations in classical electro-dynamics, and the Schrödinger wave equation in quantum mechanics. We discuss herethe properties of these functions common to all these physical situations.

The Legendre equation itself is

(1 – x2

)P′′n(x) – 2xP

′n(x) + n(n + 1)Pn(x) = 0 (G.1)

where n = 0, 1, 2, . . .. In most physics problems x = cos θ , and the Legendre equationtakes the form

1sin θ

ddθ

(sin θ

dPn(cos θ)dθ

)+ n(n + 1)Pn(cos θ) = 0 (G.2)

or more explicitly, carrying out the derivative,

d2Pn(cos θ)dθ

+ cot θdPn(cos θ)

dθ+ n(n + 1)Pn(cos θ) = 0 (G.3)

where the derivative is with respect to the polar coordinate θ rather than x. We havealready encountered this latter form in Equations 2.101, 2.102 and 2.103 when studyingthe angular solutions of Laplace’s equation.

G.1 Generating function

The Legendre polynomials can always be obtained from a generating function:

g(t, x) = (1 – 2xt + t2)–1/2 =∞∑n=0

Pn(x)tn, |t| < 1 (G.4)

Appendix G: Properties of the Legendre Functions 405

The left-hand side of this equation can be cast into a power series in tn so that the left-hand side and right-hand sides, also a power series in tn, can be equated term by term:

(1 – 2xt + t2)–1/2 =∞∑n=0

[n/2]∑k=0

(–1)k(2n – 2k)!

22n–2kn!k!(n – k)!(2x)(n–2k)tn =

∞∑n=0

Pn(x)tn (G.5)

where the upper limit on the sum, [n/2], means n/2 when n is even and (n – 1)/2 whenn is odd. Therefore, we can obtain an expression for each Pn(x) in terms of a sum overthe index k:

Pn(x) =[n/2]∑k=0

(–1)k(2n – 2k)!

2nk!(n – k)!(n – 2k)!xn–2k (G.6)

Using the EquationG.6 function we find, for example,

P0(x) = 1 P1(x) = x P2(x) =12

(3x2 – 1

)P3(x) =

52

(x3 – 3x

)

G.2 Recurrence relations

Families of functions that are solutions to a differential equation often exhibit recurrencerelations that express how a given solution Pn(x) is related in some simple way to itsneighbours, Pn+1(x) and Pn–1(x). These recurrence relations are often another, simplerway to spawn all the needed members of the family once one member is known. In thecase of the Legendre functions we can find a recurrence relation by starting from thegenerating function, EquationG.4, and differentiating it:

∂g(t, x)∂t

=x – t(

1 – 2xt + t2)3/2 =

∞∑n=0

nPn(x)tn–1 (G.7)

Using EquationG.4 we can write this last expression as the sum of four terms, each ofwhich is a power series in t:

(1 – 2xt + t2

) ∞∑n=0

nPn(x)tn–1 + (t – x)∞∑n=0

Pn(x)tn = 0 (G.8)

or

∞∑n=0

nPn(x)tn–1 –∞∑n=0

x(2n + 1)Pn(x)tn +∞∑n=0

(n + 1)Pn(x)tn+1 = 0 (G.9)

406 Appendix G: Properties of the Legendre Functions

Now we seek to rewrite the various terms so that they are all sums of a power series intn. We can do this by adjusting the indices n. Thus,

∞∑n=0

nPn(x)tn–1 is equivalent to∞∑n=0

(n + 1)Pn+1tn (G.10)

and

∞∑n=0

(n + 1)Pn(x)tn+1 is equivalent to∞∑n=0

nPn–1(x)tn (G.11)

and therefore we can write

(2n + 1)xPn(x) = (n + 1)Pn+1(x) + nPn–1(x) (G.12)

Equation G.12 is the recurrence relation for Legendre polynomials. It is a prescriptionfor expressing a given Pn(x) in terms of the adjacent polynomials in the series, Pn–1(x)and Pn+1(x). For example, it is easy to remember the first two Legendre polynomials,P0(x) = 1 and P1 = x. Using EquationG.12 we find immediately P2 = 1/2(3x2–1). Then,with P1(x) and P2(x) in hand, we find P3(x) = 1/2(5x3 – 3x). Therefore, in principle,any member of the series may be found from the recurrence relation.

G.3 Parity

It appears by inspection that even-index Legendre polynomials are even and odd-indexpolynomials are odd. In general, the parity of the polynomials can be demonstrated byrecourse to the generating function, EquationG.4:

g(t, x) = g(–t, –x)

∞∑n=0

Pn(x)tn =∞∑n=0

Pn(–x)(–t)n =∞∑n=0

Pn(–x)(–1)ntn

Equating equal powers of tn we have

Pn(x) = (–1)nPn(–x) (G.13)

which shows that, in general, even-index Legendre polynomials are even and odd-indexpolynomials are odd in the x→ –x parity operation.

Appendix G: Properties of the Legendre Functions 407

G.4 Orthogonality and normalisation

The Legendre equation, EquationG.1, can be regarded as an operator-eigenfunction-eigenvalue equation, Oψ – λψ = 0, where O the operator is

O =ddx

(1 – x2)ddx

(G.14)

The eigenfunctions ψ are the Legendre polynomials Pn(x) and the eigenvalues λ= n(n+1). Viewed in this way, it can be shown that the ‘operator’ is hermitian, and thereforethe eigenfunctions are orthogonal and the eigenvalues real. Therefore, the Legendrepolynomials are orthogonal:

∫ 1

–1Pn(x)Pm(x) dx = 0 m �= n (G.15)

and in fact they form a complete set, spanning the space of x from -1 to +1. Thenormalisation factor comes from the determination of the integral when m = n. We startwith the generating function, square it, and write

(1 – 2tx + t2)–1 =

[ ∞∑n=0

Pn(x)tn]2

(G.16)

Then we integrate both sides and note that all the PnPm cross terms will vanish by virtueof EquationG.15:

∫ 1

–1

dx1 – 2tx + t2

=∞∑n=0

∫ 1

–1[Pn(x)]2 dx t2n (G.17)

The integral on the left can be readily evaluated by making a change of variable, y =1 + 2t + t2 and dy = –2tdx. Making the substitution into EquationG.17, together withthe change in limits x = –1→ y = (1 + t)2 and x = 1→ y = (1 – t)2, we have

∫ 1

–1

dx1 – 2tx + t2

=1tln(1 + t1 – t

)=∞∑n=0

∫ 1

–1[Pn(x)]2 dx t2n (G.18)

Now, the ln expression can be expanded in a power series,

1tln(1 + t1 – t

)= 2

∞∑n=0

t2n

2n + 1(G.19)

408 Appendix G: Properties of the Legendre Functions

and therefore,

2∞∑n=0

t2n

2n + 1=∞∑n=0

∫ 1

–1[Pn(x)]2 dx t2n (G.20)

Finally, equating each term in the power series t2n on each side of EquationG.20, wehave ∫ 1

–1[Pn(x)]2 dx =

22n + 1

(G.21)

which is the normalisation condition.Using the conventional expression that summarises orthonormality, we write

∫ 1

–1Pm(x)Pn(x) dx =

2δmn2n + 1

(G.22)

Appendix HProperties of the Hermite Polynomials

The Hermite polynomials constitute the essential part of the solutions to the Schrö-dinger equation for the one-dimensional harmonic oscillator. Their properties are thestarting point for interpreting vibrational motion and dipole transition selection rulesin diatomic molecules. The Hermite polynomials can be defined from the ‘Rodriguesrepresentation’:

Hn(x) = (–1)n(exp x2

) dndxn

[exp

(–x2

)](H.1)

or by use of the generating function.

H.1 Generating function and recurrence relations

The generating function is given by

g(x, t) = e–t2+2tx =

∞∑n=0

Hn(x)tn

n!(H.2)

from which we can obtain the recurrence relations similarly to the procedure followed forthe Legendre polynomials in AppendixG. First we take the partial derivative of g(x, t)with respect to t,

∂g(x, t)∂t

= 2(x – t)e–t2+2tx =

∞∑n=0

Hn(x)ntn

n!=∞∑n=0

Hn(x)tn–1

(n – 1)!(H.3)

From which we find, after adjusting the summation indices,

– 2n∞∑n=0

Hn–1(x)tn

n!+ 2x

∞∑n=0

Hn(x)tn

n!–∞∑n=0

Hn+1(x)tn

n!= 0 (H.4)

Terms of equal powers of tn must be equal and the recurrence relation is, therefore,

Hn+1(x) = 2xHn(x) – 2nHn–1(x) (H.5)

We can also get a recurrence relation involving the derivative of the Hermite polynomialsby taking the partial derivative of the generating function with respect to x:

410 Appendix H: Properties of the Hermite Polynomials

Table H.1 Hermite polynomials.

Index n Function Polynomial

0 H0(x) 1

1 H1(x) 2x

2 H2(x) 4x2 – 2

3 H3(x) 8x3 – 12x

4 H4(x) 16x4 – 48x2 + 12

∂g(x, t)∂x

= 2te–t2+2tx = 2t

∞∑n=0

Hn(x)tn

n!=∞∑n=0

H ′n(x)tn

n!(H.6)

Then

2∞∑n=0

Hn(x)tn+1

n!= 2n

∞∑n=0

Hn–1(x)tn

n!=∞∑n=0

H ′n(x)tn

n!(H.7)

and

2nHn–1(x) = H ′n(x) (H.8)

We can get the first two members of the Hermite polynomials by expanding thegenerating function exponential in a Taylor series,

e–t2+2tx =

∞∑n=0

(2tx – t2

)nn!

= 1 + (2tx – t2) + . . . (H.9)

that results in H0(x) = 1 and H1(x) = 2x. Then from these, and EquationH.5, all thepolynomials up to arbitrary n can be calculated. The first few Hermite polynomials areshown in TableH.1.

H.2 Orthogonality and normalisation

H.2.1 Orthogonality

The Hermite polynomials themselves are not orthogonal in the sense that

∫ ∞–∞

Hn(x)Hm(x) dx �= 0 m �= n (H.10)

Appendix H: Properties of the Hermite Polynomials 411

but a closely related integral is orthogonal,

∫ ∞–∞

Hn(x)Hm(x) exp(–x2

)dx = 0 m �= n (H.11)

In fact, we can define a new orthogonal function,

ϕn(x) = exp(–x2/2)Hn(x) (H.12)

Now using the two recurrence relations, we can obtain a second order differential equa-tion in the Hermite polynomials of index n in the following way. First, by substitutingEquationH.8 into EquationH.5 we eliminate the n – 1 index,

Hn+1(x) = 2xHn(x) –H ′n(x) (H.13)

Then differentiate the result,

H ′n+1(x) = 2Hn(x) + 2xH ′n –H′′n (x) (H.14)

Use EquationH.8 again to eliminate the term with the n + 1 index,

H ′n+1(x) = 2(n + 1)Hn(x) (H.15)

Now regroup the terms in n,

H ′′n (x) – 2xH′n(x) + 2nHn(x) = 0 (H.16)

Differentiating EquationH.12 and substituting into EquationH.16 results in

ϕ′′n(x) + (2n + 1 – x2)ϕn(x) = 0 (H.17)

which is the equation for the one-dimensional quantummechanical oscillator. Therefore,we see that the physically significant entities are not the Hermite polynomials per se butthe orthogonal functions, ϕn(x) = Hn(x) exp(–x2/2).

H.2.2 Normalisation

We find the normalisation of the Hermite functions, EquationH.12, by multiplying thegenerating function for Hm(x) with that for Hn(x), similar to the approach we take withthe squared generating function of the Legendre polynomials, EquationG.16,

exp(–s2 + 2sx) exp(–t2 + 2tx) =∞∑

m,n=0

Hm(x)Hn(x)smtn

m!n!(H.18)

412 Appendix H: Properties of the Hermite Polynomials

Then multiply both sides by exp(–x2) and integrate over the interval from –∞ to∞. Cross terms in the exp(–x2)Hm(x)Hn(x) products drop out of the integral due toorthogonality:

∞∑n=0

(st)n

n!n!

∫ ∞–∞

exp(–x2) [Hn(x)]2 dx =∫ ∞–∞

exp –[(x – s – t)2] exp(2st) dx (H.19)

The first exponential term under the integral on the right-hand side is just the errorfunction, and the integral over it is known to be

√π . The remaining term, exp(2st), can

be expanded in a Taylor series and we have, therefore,

∞∑n=0

(st)n

n!n!

∫ ∞–∞

exp(–x2) [Hn(x)]2 dx =√π

∞∑n=0

2n(st)n

n!(H.20)

Now in the usual way we equate equal powers of (st)n that results in∫ ∞–∞

exp(–x2) [Hn(x)]2 dx = 2n√πn! (H.21)

Equation H.21 is the normalisation expression for the Hermite functions.

Index

AABCD matrix 194, 225ABCD network 197Abraham momentum 261,

262Abraham momentum,

dipole 296Abraham-Minkowski

controversy 264absorption coefficient 45admittance matrix 190Ag-TiO2 238, 243Alhazen 2Ampère’s law 14, 372Amperian current loop 267amperian current loop 283angular equation 338angular momentum 342anisotropy 250anisotropy in stacked

layers 242antenna, half-wave 103antenna, real 102Arago spot 7array, antenna dipole

half-wave 104associated Laguerre

polynomials 336associated Legendre

functions 338atom cooling 301atom interferometer 286atom recoil 286atomic absorption,

emission 352atomic spectroscopy 328atomic transitions 350atomism 1azimuthal functions 338

BB-field 14B-field wave equation 50Balazs thought experiment 259band edge 231band gap 227, 232

Bessel functions 185Biot-Savart law 14blackbody radiation 108Bloch vector 237, 247Bloch wave 226Bloch wave dispersion 227Bloch waves 225, 246Bohr correspondence

principle 332Born-Oppenheimer

approximation 359Bose-Einstein condensate 286bound charge density 17bound current 267bound current density 88boundary conditions 176, 185Boyle, Robert 7Bragg reflection 230broadband excitation 310bulk plasmon frequency 83bulk plasmon resonance 131

Ccanonical momentum 290capacitance 157Cartesian coordinates 334cavity modes 108charge conservation 268circuit theory 154Clausius-Mossotti relation 98conduction current 132conductivity 53, 56, 57constants of separation 335constitutive relations 15continuity conditions 31, 72,

218continuity equation 256continuity relations 135correspondence principle 332,

362Coulomb potential 334Coulomb’s law 13, 370cross section 305, 331current 164current density 80, 83curvilinear coordinates 388

curvilinear coordinates,curl 392

curvilinear coordinates,divergence 391

curvilinear coordinates,gradient 391

curvilinear coordinates,Laplacian 392

curvilinear coordinates, Stokes’theorem 392

cutoff parameter 177cutoff propagationparameter 180

cylindrical coordinates 393

DD-field 15Dalton, John 8damped harmonicoscillator 82, 331

damping rate 133Democritus 1Descartes, Renè 4detuning convention 314dielectric 72dielectric constant 44, 123,127

dielectric periodic layers 235dielectric sphere 200dielectric-metal periodiclayers 235

diffraction 5dipole 241dipole antenna 100dipole emission 328dipole matrix element 357dipole radiated power 101dipole radiation 328, 356dipole radiative impedance 102dipole transition moment 311dipole transitions 328dipole-gradient force 302, 316,320

Dirac notation 279directional gain 102dispersion 88

414 Index

dispersion equation 243dispersion relation 95, 126,129

displacement current 18, 200,201

displacement field 14, 42, 97,122

dissipated energy 89dissipated energy density

92, 93dissipative loss 312divergence operator 382Doppler cooling 325Doppler cooling limit 325Drude model 127Drude-Lorentz dispersion 88dyadic 278

EE-field 14E-field wave equation 50effective medium theory 239,245

eigen energy 342, 362eigenstate 332Einstein A, B coefficients 113,

354Einstein Box experiment 257Einstein rate equation 312Einstein, Albert 9Einstein-Laub force law 280,

284electric dipole 302, 328electric field 42electromagnetic system of

units 372electron 303, 328electronic states 363Empedocles 1EMT 239, 245, 252energy conservation 269energy density 44, 57energy flow 86energy flux 45, 57, 71, 134,

138, 215ENZ 239, 240epsilon-near-zero 239equipartition of energy 111equivalent circuit 154, 198,

202, 207equivalent conductivity 56esu system of units 373Euclid 2evanescent wave 121

FFaraday’s law 58Faraday’s law of induction 372Faraday, Michael 7FDTD 233FDTD simulation 241Fermat, Pierre 5free charge density 17free-electron gas 82free-electron plasma 126Fresnel coefficients 219, 221,

249Fresnel laws 218Fresnel relations 64, 68, 85Fresnel, Augustin-Jean 6, 7

Ggap plasmons 138Gaussian system of units 373Gay-Lussac, Joseph Louis 8good conductor 56, 74grad, div, curl in cylindrical

coordinates 393grad, div, curl in polar

coordinates 393gradient operator 382Grimaldi 5Grotrian diagram 352guided waves 138

HH-field 15Hamiltonian 332Hamiltonian operator 333harmonic oscillator 303, 329harmonic oscillator model 361harmonic oscillator,

damped 302, 305Heaviside-Lorentz system of

units 372Helmholtz equation 53, 164,

175Hermite function

normalization 410Hermite functions 411Hermite polynomial generating

function 409Hermite polynomial

orthogonality 410Hermite polynomial recurrence

relation 409Hermite polynomial recursion

relation 409Hermite polynomials 409, 411

Hertz vector 116hidden energy 270, 282hidden momentum 264, 282Hooke, Robert 5Huygens, Christiaan 6hydrogen atom 333hydrogen radial equation 336

IIMI waveguides 139impedance 53, 55, 57, 72, 77,162, 170, 188, 201–203, 250

impedance matrix 190impedance network 197index of refraction 5, 44, 51,82, 92, 123, 125

induction 157inductive transformer 196intrinsic spin 358

JJoule heating 80, 257, 269

KKapitza-Diracinterferometer 287

kinetic energy 332kinetic momentum 290Kirchhoff ’s current rule 158Kirchhoff ’s rules 155Kronicker delta 278

LLagrange bracket 290Laguerre polynomialgenerating function 400

Laguerre polynomialnormalization 401

Laguerre polynomialorthogonality 401

Laguerre polynomial recursionrelation 400

Laguerre polynomials 336,400

Laguerre polynomials,associated 402

Laplacian operator 334Lavoisier, Antoine 8left-handed materials 211Legendre functions 338, 404Legendre orthogonalfunctions 407

Legendre polynomialgenerating function 404

Index 415

Legendre polynomialrecurrence relation 405

Leucippus 1light line 132line junction 167Lorentz field momentum 279Lorentz force 262Lorentz Force Law 14Lorentz force law 279Lorentz model 265, 267lossy 72lossy transmission line 170Lucretius 1lumped circuit elements 155

Mmagnetic dipole model 268magnetic field 14, 374magnetic induction field 51,

372, 374magnetisation field 16magnetization 374Mansuripur 264matrix element 308matrix elements 348Maxwell stress tensor 271, 280Maxwell’s equations 17Maxwell, James Clerk 7Maxwell-Ampère law 58Maxwell-Boltzmann

constant 357Meissner effect 241Mendeleev, Dimitri 8metal-dielectric periodic

layers 233metallic sphere 202metamaterials 211Michelson-Morley

experiment 9MIM waveguide 139, 213Minkowski momentum 264,

289Minkowski momentum,

dipole 296Minkowsky momentum 261,

262mirrors, immersed 284MKS system of units 371molecular spectroscopy 361moment of inertia 359momentum 333momentum in EM fields 257momentum operator 333

Nnanosphere 199nanostructures 198negative permittivity 213negative refractive index 211,

213networks 189Newton, Isaac 7normalisation 337

OOBE 319optical Bloch equations 319optical trap 320orthogonal matrix 274orthogonal transformation

275, 277orthonormal wave

functions 341

Pparallel-plate waveguide 175parity operation 351PEC 173perfect conductor 75perfect metal 75periodic stacked layers 229permeability 14, 15permittivity 14, 15, 42phase velocity 132, 164phasor addition 396phasors 49, 57–59, 173, 396phasors and circuit

analysis 397photonic band gaps 226Pi-network 195Planck distribution 111, 113Planck relation 347plane wave 48, 55, 69, 81,

164, 168, 202plasma frequency 133polar coordinates 393polar functions 338polarisability 98polarisation 59polarisation field 16polarization 86, 127, 128potential energy 332power density 80Poynting vector 59, 71, 74,

134Poynting’s theorem 86Priestly, Joseph 8

principal quantumnumber 352

propagation parameter 44, 49,52, 70, 82, 141, 164, 171,186

propagation vector 48Ptolemy 2

Qquantum numbers 339quarter-wave transformer 170quasistatic approximation 154

RRabi frequency 294, 308radial equation 335radiated power dipole 241radiation pressure force 301,316, 318

radiative impedance 104Rayleigh–Jeans law 111real orbitals 345reciprocal networks 197reflection 69, 165, 167, 229reflection coefficient 71,72, 77

reflection thoughtexperiment 262

reflectivity 249resistance 156resonant tunnelling 231, 239rigid rotor 358Rodrigues form, Laguerrepolynomials 401

Rodrigues representation 409rotating waveapproximation 310

RWA 310, 314

SS-parameters 248, 249scattering cross section 305,331

Schrödinger equation 332selection rules 328, 350, 352semiclassical absorption,emission 346

separation of coordinates 335Shockley, James 284skin depth 57, 77sodium atom 304, 330Sommerfeld, Arnold 116spectral line width 353spherical coordinates 334

416 Index

spherical harmonicfunctions 339

spontaneous emission 112,114, 301, 311

spontaneous emissionrate 311, 355

spontaneous force 301spp 117, 138spp waves 235Stokes’ theorem 18, 59, 159,

162, 386stop bands 226stopband 132stored energy 89, 90stored energy density 91, 93stress tensor 271subwavelength 154surface charge 122surface charge density 97surface plasmon 116, 138surface plasmon resonance 131surface plasmons 233surface wave 115, 125, 142susceptibility 16, 42, 88, 96,

313

TT-circuit 207T-network 195, 197TE modes 175, 183TE polarization 117, 147tensor calculus 272tensor, second rank 271, 272TM modes 177, 188TM polarisation 117TM polarization 138TMT 225, 229, 232, 252Torricelli, Evangelista 6transcendental equation 144,

209transfer matrix theory 246transition moment 311transmission 69, 167, 229, 248transmission coefficient 71,

72, 77transmission line 159, 162,

168, 169, 202transmittance 231, 232transpose, matrix 274transverse resonance 206two-level atom 306

Uultraviolet catastrophe 111uncertainty principle 344

uniaxial anisotropic 243unit vectors 375units 369

Vvector calculus 375vector cross product 379vector curl operation 384vector multiplication 378vector rotation 375vector scalar product 378vector triple product 381vectors 375vibrational transitions 363voltage 164

Wwave function 332waveguide 154, 172waveguides, cylindrical 182waveguides-2D 178

YYoung, Thomas 6, 7

ZZenneck waves 116

AUTHORS

JohnWeiner graduated fromMrs.Warnock’s kindergarten on Hathaway Lane in Haver-town, Pennsylvania, with a dual major in finger painting and rhythm sticks, in 1949. Hewas subsequently promoted from Mrs. Warnock’s to the adjacent Oakmont Elemen-tary School. In the 5th and 6th grades he played right tackle, left end, and left halfback(T-formation) on the football team, graduating in June of 1955. He advanced to theHaverford Junior High School in September, but he was too small to play on the footballteam as a 7th grader. Instead he joined the marching band (trombone and baritone); andbecause the trombones always marched in the first rank (to leave enough space for theirslides) and right behind the majorettes, he had occasion to fall in love with the head drummajorette, marching right in front of him. His love was unrequited (she was a 9th graderand the most popular girl in the high school). Despite this disappointment he managedto pass the 7th, 8th, and 9th grades. Regretfully leaving his majorette heart-throb atHaverford High, he and his family in the summer of 1958 moved to Chambersburg,Pennsylvania, where he enrolled in the Chambersburg Area Senior High School. Hefinally made the junior varsity (JV) football squad but was still too small for the var-sity. Nevertheless during the JV game between Chambersburg and Scotland School hecaught a 40 yard pass from quarterback Kirby Keller and ran for the only touchdownthat the team was to score that day. It was a unique moment of unalloyed happiness. Inthe summer of 1960 his family moved again to Bethesda, Maryland, where he graduatedfrom high school in June of 1961. From 1961 to 1964 he attended Penn State in thetown of State College, leaving in June of 1964 with a BS degree in Chemistry (at PennState playing football was out of the question). From 1964 to 1970 he attended graduateschool at the University of Chicago, earning a PhD in Chemical Physics in June of 1970.After a post-doctoral interlude at Yale he got his first ‘real’ job as an assistant professorat Dartmouth College in September of 1973. In 1978 he was invited to the Universitéde Paris for a year, and while there accepted an offer from the University of Maryland,College Park.

At Maryland Weiner established a research group studying the collisional process,associative ionisation. Sodium atoms were a convenient experimental choice for thesestudies, and he became interested in ‘laser-induced’ collisions – the idea of collidingatoms or ions absorbing a light quantum to produce chemical binding. Coincidently,at the National Institute of Standards and Technology (NIST), W. D. Phillips’ groupwas learning how to cool a beam of sodium atoms to submilliKelvin temperatures.Using cold atoms to study extremely slow associative ionisation collisions was an ob-vious consequence, and Weiner spent the next ten years focused on various aspects of‘ultracold collisions’. In 1997 he joined the faculty at the Université Paul Sabatier in Tou-louse, France, where his interests shifted to how nanoscale structures might be used tomanipulate cooled and trapped atom clouds and condensed quantum gases. In 2006 he

418 Authors

retired from his post in Toulouse becoming a visiting researcher at the Instituto de Físicade S ao Carlos (IFSC) in the Universidade de S ao Paulo, Brazil. In the years 2009-2010he was a visiting fellow at the Center for Nanoscale Science and Technology (CNST) atNIST, Gaithersburg before returning to the IFSC for another and final year. Weiner iscurrently living in Paris, France, where he is tolerated by Samba and Annick, his cat andwife, respectively. He reads a lot, attempts to think clearly about some difficult subjectsin the morning, and tries not to make stupid mistakes for the rest of the day. The successor failure of this latter effort will be judged by readers of this book.

Frederico Nunes was born in Recife, Pernambuco State, Brazil, in 1947. He obtainedan undergraduate degree in electrical engineering from the School of Engineering at theFederal University of Pernambuco (UFPE) in 1971. He continued his advanced edu-cation by enrolling in the Physics and Pure Mathematics curriculum at the Institute ofProfessor GlebWataghin, State University of Campinas (UNICAMP), obtaining his MSand PhD degrees in 1974 and 1977, respectively. Since then he has been on the faculty atseveral Brazilian universities and since 1997, he has been an associate professor at UFPEin Recife. He has received the Marechal Trompowski award for excellence in teaching atUFPE and has made many research contributions in the area of optics, semiconductors,photonics, and nanoplasmonics. He has authored many scientific publications and holdsa number of patents in the technologies associated with these scientific disciplines.

John Weiner Frederico Nunes

All Oakmont Elementary School right tackle,1954. ‘Always fight for Brown and Whiteand march to vic-tor-y’.

light-matter interaction : physics and engineering at the nanoscale

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